(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 110626, 3888]*) (*NotebookOutlinePosition[ 111364, 3914]*) (* CellTagsIndexPosition[ 111320, 3910]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ " Sigma ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[Sum]\)\)\)]], " , Riemann Sums\n Definite Integrals" }], "Subtitle", TextAlignment->Left, TextJustification->0], Cell[TextData[{ " ", StyleBox["Mathematica", FontSlant->"Italic"], " " }], "Subtitle", TextAlignment->Left, TextJustification->0], Cell["\<\ Dr. Andrzej Korzeniowski Department of Mathematics, UTA\ \>", "Subsubtitle"] }, Open ]], Cell[CellGroupData[{ Cell["Lesson 5 ", "Subtitle"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " built-in symbolic summation is a great tool for finding Riemann sums of \ elementary functions. \nBelow are several examples." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Sum]\+\(i = 1\)\%\[Infinity] 1\/i\^12\)], "Input"], Cell[BoxData[ \(\(691\ \[Pi]\^12\)\/638512875\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Sum]\+\(i = 1\)\%n i\^12\)], "Input"], Cell[BoxData[ \(\(1\/2730\) \((n\ \((1 + n)\)\ \((1 + 2\ n)\)\ \((\(-691\) + 2073\ n - 287\ n\^2 - 3285\ n\^3 + 1420\ n\^4 + 2310\ n\^5 - 1190\ n\^6 - 1050\ n\^7 + 525\ n\^8 + 525\ n\^9 + 105\ n\^10)\))\)\)], "Output"] }, Open ]], Cell["", "Input"], Cell[BoxData[ \(\(\(<< Graphics`FilledPlot`\)\(\ \ \)\)\)], "Input"], Cell[BoxData[ \(\(\( (*\ Hue[c]\ \ or\ GrayLevel[c], \ 0\ \[LessEqual] \ c\ \[LessEqual] \ 1\ give\ different\ colors . \ \ \[IndentingNewLine]If\ \(Options : Fills\) \[Rule] \ is\ not\ used\ then\ default\ color\ is\ used\ *) \)\(\ \[IndentingNewLine]\)\)\)], "Input", CellMargins->{{23, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(FilledPlot[{x\^4 - 1, 1 - x\^2}, {x, \(-1\), 2}];\)\(\ \)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 0.618034 [ [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.34127 0.31746 0.267426 0.0842371 [ [.02381 .25493 -6 -9 ] [.02381 .25493 6 0 ] [.18254 .25493 -12 -9 ] [.18254 .25493 12 0 ] [.5 .25493 -9 -9 ] [.5 .25493 9 0 ] [.65873 .25493 -3 -9 ] [.65873 .25493 3 0 ] [.81746 .25493 -9 -9 ] [.81746 .25493 9 0 ] [.97619 .25493 -3 -9 ] [.97619 .25493 3 0 ] [.32877 .01472 -12 -4.5 ] [.32877 .01472 0 4.5 ] [.32877 .09895 -12 -4.5 ] [.32877 .09895 0 4.5 ] [.32877 .18319 -12 -4.5 ] [.32877 .18319 0 4.5 ] [.32877 .35166 -6 -4.5 ] [.32877 .35166 0 4.5 ] [.32877 .4359 -6 -4.5 ] [.32877 .4359 0 4.5 ] [.32877 .52014 -6 -4.5 ] [.32877 .52014 0 4.5 ] [.32877 .60437 -6 -4.5 ] [.32877 .60437 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .26743 m .02381 .27368 L s [(-1)] .02381 .25493 0 1 Mshowa .18254 .26743 m .18254 .27368 L s [(-0.5)] .18254 .25493 0 1 Mshowa .5 .26743 m .5 .27368 L s [(0.5)] .5 .25493 0 1 Mshowa .65873 .26743 m .65873 .27368 L s [(1)] .65873 .25493 0 1 Mshowa .81746 .26743 m .81746 .27368 L s [(1.5)] .81746 .25493 0 1 Mshowa .97619 .26743 m .97619 .27368 L s [(2)] .97619 .25493 0 1 Mshowa .125 Mabswid .05556 .26743 m .05556 .27118 L s .0873 .26743 m .0873 .27118 L s .11905 .26743 m .11905 .27118 L s .15079 .26743 m .15079 .27118 L s .21429 .26743 m .21429 .27118 L s .24603 .26743 m .24603 .27118 L s .27778 .26743 m .27778 .27118 L s .30952 .26743 m .30952 .27118 L s .37302 .26743 m .37302 .27118 L s .40476 .26743 m .40476 .27118 L s .43651 .26743 m .43651 .27118 L s .46825 .26743 m .46825 .27118 L s .53175 .26743 m .53175 .27118 L s .56349 .26743 m .56349 .27118 L s .59524 .26743 m .59524 .27118 L s .62698 .26743 m .62698 .27118 L s .69048 .26743 m .69048 .27118 L s .72222 .26743 m .72222 .27118 L s .75397 .26743 m .75397 .27118 L s .78571 .26743 m .78571 .27118 L s .84921 .26743 m .84921 .27118 L s .88095 .26743 m .88095 .27118 L s .9127 .26743 m .9127 .27118 L s .94444 .26743 m .94444 .27118 L s .25 Mabswid 0 .26743 m 1 .26743 L s .34127 .01472 m .34752 .01472 L s [(-3)] .32877 .01472 1 0 Mshowa .34127 .09895 m .34752 .09895 L s [(-2)] .32877 .09895 1 0 Mshowa .34127 .18319 m .34752 .18319 L s [(-1)] .32877 .18319 1 0 Mshowa .34127 .35166 m .34752 .35166 L s [(1)] .32877 .35166 1 0 Mshowa .34127 .4359 m .34752 .4359 L s [(2)] .32877 .4359 1 0 Mshowa .34127 .52014 m .34752 .52014 L s [(3)] .32877 .52014 1 0 Mshowa .34127 .60437 m .34752 .60437 L s [(4)] .32877 .60437 1 0 Mshowa .125 Mabswid .34127 .03156 m .34502 .03156 L s .34127 .04841 m .34502 .04841 L s .34127 .06526 m .34502 .06526 L s .34127 .0821 m .34502 .0821 L s .34127 .1158 m .34502 .1158 L s .34127 .13265 m .34502 .13265 L s .34127 .14949 m .34502 .14949 L s .34127 .16634 m .34502 .16634 L s .34127 .20004 m .34502 .20004 L s .34127 .21688 m .34502 .21688 L s .34127 .23373 m .34502 .23373 L s .34127 .25058 m .34502 .25058 L s .34127 .28427 m .34502 .28427 L s .34127 .30112 m .34502 .30112 L s .34127 .31797 m .34502 .31797 L s .34127 .33482 m .34502 .33482 L s .34127 .36851 m .34502 .36851 L s .34127 .38536 m .34502 .38536 L s .34127 .40221 m .34502 .40221 L s .34127 .41905 m .34502 .41905 L s .34127 .45275 m .34502 .45275 L s .34127 .4696 m .34502 .4696 L s .34127 .48644 m .34502 .48644 L s .34127 .50329 m .34502 .50329 L s .34127 .53699 m .34502 .53699 L s .34127 .55383 m .34502 .55383 L s .34127 .57068 m .34502 .57068 L s .34127 .58753 m .34502 .58753 L s .25 Mabswid .34127 0 m .34127 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .26743 m .04262 .24917 L .06244 .23332 L .08426 .21938 L .10458 .20922 L .12297 .20203 L .14264 .1961 L .16394 .19139 L .17372 .18973 L .18408 .18825 L .19359 .18713 L .20406 .18613 L .21351 .1854 L .2225 .18484 L .23263 .18434 L .2419 .184 L .25189 .18372 L .2575 .1836 L .26278 .1835 L .26774 .18343 L .2731 .18337 L .27756 .18333 L .28246 .18329 L .28777 .18326 L .29338 .18323 L .29589 .18322 L .29858 .18322 L .30343 .18321 L .30599 .1832 L .30871 .1832 L .31104 .1832 L .3136 .18319 L .31635 .18319 L .31779 .18319 L .31931 .18319 L .32065 .18319 L .32211 .18319 L .32334 .18319 L .32467 .18319 L .32603 .18319 L .32731 .18319 L .32856 .18319 L .32971 .18319 L .33095 .18319 L .33166 .18319 L .33232 .18319 L .33301 .18319 L .33366 .18319 L .33441 .18319 L .33509 .18319 L Mistroke .33626 .18319 L .3375 .18319 L .33867 .18319 L .33973 .18319 L .34098 .18319 L .34169 .18319 L .34234 .18319 L .34353 .18319 L .34479 .18319 L .34601 .18319 L .34671 .18319 L .34735 .18319 L .34852 .18319 L .34976 .18319 L .35088 .18319 L .35192 .18319 L .35306 .18319 L .35429 .18319 L .35553 .18319 L .35688 .18319 L .35829 .18319 L .3596 .18319 L .36196 .18319 L .36327 .18319 L .36447 .18319 L .36683 .18319 L .369 .18319 L .37139 .1832 L .37391 .1832 L .37815 .1832 L .38056 .18321 L .38281 .18321 L .38781 .18323 L .39325 .18325 L .39838 .18328 L .40314 .18331 L .41213 .1834 L .41667 .18346 L .42169 .18354 L .43177 .18375 L .44101 .18401 L .45116 .1844 L .46186 .18494 L .47139 .18557 L .48158 .1864 L .49994 .18845 L .51016 .18994 L .52126 .19189 L .5414 .19649 L .56125 .20261 L Mistroke .58229 .21118 L .60184 .22142 L .62014 .23335 L .65894 .26764 L .68009 .29248 L .70018 .32081 L .73839 .38946 L .77906 .48784 L .81821 .61232 L Mfstroke .81821 .61232 m .8196 .61803 L s .02381 .26743 m .06244 .28668 L .10458 .30484 L .14415 .31919 L .18221 .33052 L .22272 .33992 L .24141 .34333 L .26171 .34637 L .27279 .34774 L .28302 .34883 L .29341 .34975 L .30316 .35045 L .31218 .35096 L .31682 .35116 L .32188 .35135 L .32474 .35144 L .3274 .3515 L .33016 .35156 L .33265 .3516 L .33383 .35162 L .33511 .35163 L .33621 .35164 L .33742 .35165 L .33871 .35166 L .33941 .35166 L .34007 .35166 L .34125 .35166 L .34251 .35166 L .34372 .35166 L .34442 .35166 L .34507 .35165 L .34638 .35164 L .34779 .35163 L .35034 .35159 L .35268 .35155 L .35807 .35143 L .36375 .35124 L .37336 .3508 L .38387 .35015 L .40348 .34843 L .42443 .34588 L .44577 .34254 L .46593 .33868 L .50439 .32942 L .54531 .31687 L .58471 .30213 L .6226 .28551 L .66294 .26518 L .70177 .24304 L .74304 .21674 L Mistroke .7828 .18872 L .82105 .15926 L .86175 .12524 L .90093 .08986 L .94256 .04946 L .97619 .01472 L Mfstroke 0 1 1 r .02381 .26743 m .04262 .24917 L .06244 .23332 L .08426 .21938 L .10458 .20922 L .12297 .20203 L .14264 .1961 L .16394 .19139 L .17372 .18973 L .18408 .18825 L .19359 .18713 L .20406 .18613 L .21351 .1854 L .2225 .18484 L .23263 .18434 L .2419 .184 L .25189 .18372 L .2575 .1836 L .26278 .1835 L .26774 .18343 L .2731 .18337 L .27756 .18333 L .28246 .18329 L .28777 .18326 L .29338 .18323 L .29589 .18322 L .29858 .18322 L .30343 .18321 L .30599 .1832 L .30871 .1832 L .31104 .1832 L .3136 .18319 L .31635 .18319 L .31779 .18319 L .31931 .18319 L .32065 .18319 L .32211 .18319 L .32334 .18319 L .32467 .18319 L .32603 .18319 L .32731 .18319 L .32856 .18319 L .32971 .18319 L .33095 .18319 L .33166 .18319 L .33232 .18319 L .33301 .18319 L .33366 .18319 L .33441 .18319 L .33509 .18319 L .33626 .18319 L .3375 .18319 L .33867 .18319 L .33973 .18319 L .34098 .18319 L .34169 .18319 L .34234 .18319 L .34353 .18319 L .34479 .18319 L .34601 .18319 L .34671 .18319 L .34735 .18319 L .34852 .18319 L .34976 .18319 L .35088 .18319 L .35192 .18319 L .35306 .18319 L .35429 .18319 L .35553 .18319 L .35688 .18319 L .35829 .18319 L .3596 .18319 L .36196 .18319 L .36327 .18319 L .36447 .18319 L .36683 .18319 L .369 .18319 L .37139 .1832 L .37391 .1832 L .37815 .1832 L .38056 .18321 L .38281 .18321 L .38781 .18323 L .39325 .18325 L .39838 .18328 L .40314 .18331 L .41213 .1834 L .41667 .18346 L .42169 .18354 L .43177 .18375 L .44101 .18401 L .45116 .1844 L .46186 .18494 L .47139 .18557 L .48158 .1864 L .49994 .18845 L .51016 .18994 L .52126 .19189 L .5414 .19649 L .56125 .20261 L .58229 .21118 L .60184 .22142 L .62014 .23335 L .65894 .26764 L .68009 .29248 L .70018 .32081 L .73839 .38946 L .77906 .48784 L .81821 .61232 L .85981 .78281 L .8999 .99086 L .93847 1.2381 L .97619 1.53098 L .97619 .01472 L .94256 .04946 L .90093 .08986 L .86175 .12524 L .82105 .15926 L .7828 .18872 L .74304 .21674 L .70177 .24304 L .66294 .26518 L .6226 .28551 L .58471 .30213 L .54531 .31687 L .50439 .32942 L .46593 .33868 L .44577 .34254 L .42443 .34588 L .40348 .34843 L .38387 .35015 L .37336 .3508 L .36375 .35124 L .35807 .35143 L .35268 .35155 L .35034 .35159 L .34779 .35163 L .34638 .35164 L .34507 .35165 L .34442 .35166 L .34372 .35166 L .34251 .35166 L .34125 .35166 L .34007 .35166 L .33941 .35166 L .33871 .35166 L .33742 .35165 L .33621 .35164 L .33511 .35163 L .33383 .35162 L .33265 .3516 L .33016 .35156 L .3274 .3515 L .32474 .35144 L .32188 .35135 L .31682 .35116 L .31218 .35096 L .30316 .35045 L .29341 .34975 L .28302 .34883 L .27279 .34774 L .26171 .34637 L .24141 .34333 L .22272 .33992 L .18221 .33052 L .14415 .31919 L .10458 .30484 L .06244 .28668 L .02381 .26743 L F 0 g .02381 .26743 m .04262 .24917 L .06244 .23332 L .08426 .21938 L .10458 .20922 L .12297 .20203 L .14264 .1961 L .16394 .19139 L .17372 .18973 L .18408 .18825 L .19359 .18713 L .20406 .18613 L .21351 .1854 L .2225 .18484 L .23263 .18434 L .2419 .184 L .25189 .18372 L .2575 .1836 L .26278 .1835 L .26774 .18343 L .2731 .18337 L .27756 .18333 L .28246 .18329 L .28777 .18326 L .29338 .18323 L .29589 .18322 L .29858 .18322 L .30343 .18321 L .30599 .1832 L .30871 .1832 L .31104 .1832 L .3136 .18319 L .31635 .18319 L .31779 .18319 L .31931 .18319 L .32065 .18319 L .32211 .18319 L .32334 .18319 L .32467 .18319 L .32603 .18319 L .32731 .18319 L .32856 .18319 L .32971 .18319 L .33095 .18319 L .33166 .18319 L .33232 .18319 L .33301 .18319 L .33366 .18319 L .33441 .18319 L .33509 .18319 L Mistroke .33626 .18319 L .3375 .18319 L .33867 .18319 L .33973 .18319 L .34098 .18319 L .34169 .18319 L .34234 .18319 L .34353 .18319 L .34479 .18319 L .34601 .18319 L .34671 .18319 L .34735 .18319 L .34852 .18319 L .34976 .18319 L .35088 .18319 L .35192 .18319 L .35306 .18319 L .35429 .18319 L .35553 .18319 L .35688 .18319 L .35829 .18319 L .3596 .18319 L .36196 .18319 L .36327 .18319 L .36447 .18319 L .36683 .18319 L .369 .18319 L .37139 .1832 L .37391 .1832 L .37815 .1832 L .38056 .18321 L .38281 .18321 L .38781 .18323 L .39325 .18325 L .39838 .18328 L .40314 .18331 L .41213 .1834 L .41667 .18346 L .42169 .18354 L .43177 .18375 L .44101 .18401 L .45116 .1844 L .46186 .18494 L .47139 .18557 L .48158 .1864 L .49994 .18845 L .51016 .18994 L .52126 .19189 L .5414 .19649 L .56125 .20261 L Mistroke .58229 .21118 L .60184 .22142 L .62014 .23335 L .65894 .26764 L .68009 .29248 L .70018 .32081 L .73839 .38946 L .77906 .48784 L .81821 .61232 L Mfstroke .81821 .61232 m .8196 .61803 L s .02381 .26743 m .06244 .28668 L .10458 .30484 L .14415 .31919 L .18221 .33052 L .22272 .33992 L .24141 .34333 L .26171 .34637 L .27279 .34774 L .28302 .34883 L .29341 .34975 L .30316 .35045 L .31218 .35096 L .31682 .35116 L .32188 .35135 L .32474 .35144 L .3274 .3515 L .33016 .35156 L .33265 .3516 L .33383 .35162 L .33511 .35163 L .33621 .35164 L .33742 .35165 L .33871 .35166 L .33941 .35166 L .34007 .35166 L .34125 .35166 L .34251 .35166 L .34372 .35166 L .34442 .35166 L .34507 .35165 L .34638 .35164 L .34779 .35163 L .35034 .35159 L .35268 .35155 L .35807 .35143 L .36375 .35124 L .37336 .3508 L .38387 .35015 L .40348 .34843 L .42443 .34588 L .44577 .34254 L .46593 .33868 L .50439 .32942 L .54531 .31687 L .58471 .30213 L .6226 .28551 L .66294 .26518 L .70177 .24304 L .74304 .21674 L Mistroke .7828 .18872 L .82105 .15926 L .86175 .12524 L .90093 .08986 L .94256 .04946 L .97619 .01472 L Mfstroke MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.34127 0.31746 0.267426 0.0842371 [ [.02381 .25493 -6 -9 ] [.02381 .25493 6 0 ] [.18254 .25493 -12 -9 ] [.18254 .25493 12 0 ] [.5 .25493 -9 -9 ] [.5 .25493 9 0 ] [.65873 .25493 -3 -9 ] [.65873 .25493 3 0 ] [.81746 .25493 -9 -9 ] [.81746 .25493 9 0 ] [.97619 .25493 -3 -9 ] [.97619 .25493 3 0 ] [.32877 .01472 -12 -4.5 ] [.32877 .01472 0 4.5 ] [.32877 .09895 -12 -4.5 ] [.32877 .09895 0 4.5 ] [.32877 .18319 -12 -4.5 ] [.32877 .18319 0 4.5 ] [.32877 .35166 -6 -4.5 ] [.32877 .35166 0 4.5 ] [.32877 .4359 -6 -4.5 ] [.32877 .4359 0 4.5 ] [.32877 .52014 -6 -4.5 ] [.32877 .52014 0 4.5 ] [.32877 .60437 -6 -4.5 ] [.32877 .60437 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .26743 m .02381 .27368 L s [(-1)] .02381 .25493 0 1 Mshowa .18254 .26743 m .18254 .27368 L s [(-0.5)] .18254 .25493 0 1 Mshowa .5 .26743 m .5 .27368 L s [(0.5)] .5 .25493 0 1 Mshowa .65873 .26743 m .65873 .27368 L s [(1)] .65873 .25493 0 1 Mshowa .81746 .26743 m .81746 .27368 L s [(1.5)] .81746 .25493 0 1 Mshowa .97619 .26743 m .97619 .27368 L s [(2)] .97619 .25493 0 1 Mshowa .125 Mabswid .05556 .26743 m .05556 .27118 L s .0873 .26743 m .0873 .27118 L s .11905 .26743 m .11905 .27118 L s .15079 .26743 m .15079 .27118 L s .21429 .26743 m .21429 .27118 L s .24603 .26743 m .24603 .27118 L s .27778 .26743 m .27778 .27118 L s .30952 .26743 m .30952 .27118 L s .37302 .26743 m .37302 .27118 L s .40476 .26743 m .40476 .27118 L s .43651 .26743 m .43651 .27118 L s .46825 .26743 m .46825 .27118 L s .53175 .26743 m .53175 .27118 L s .56349 .26743 m .56349 .27118 L s .59524 .26743 m .59524 .27118 L s .62698 .26743 m .62698 .27118 L s .69048 .26743 m .69048 .27118 L s .72222 .26743 m .72222 .27118 L s .75397 .26743 m .75397 .27118 L s .78571 .26743 m .78571 .27118 L s .84921 .26743 m .84921 .27118 L s .88095 .26743 m .88095 .27118 L s .9127 .26743 m .9127 .27118 L s .94444 .26743 m .94444 .27118 L s .25 Mabswid 0 .26743 m 1 .26743 L s .34127 .01472 m .34752 .01472 L s [(-3)] .32877 .01472 1 0 Mshowa .34127 .09895 m .34752 .09895 L s [(-2)] .32877 .09895 1 0 Mshowa .34127 .18319 m .34752 .18319 L s [(-1)] .32877 .18319 1 0 Mshowa .34127 .35166 m .34752 .35166 L s [(1)] .32877 .35166 1 0 Mshowa .34127 .4359 m .34752 .4359 L s [(2)] .32877 .4359 1 0 Mshowa .34127 .52014 m .34752 .52014 L s [(3)] .32877 .52014 1 0 Mshowa .34127 .60437 m .34752 .60437 L s [(4)] .32877 .60437 1 0 Mshowa .125 Mabswid .34127 .03156 m .34502 .03156 L s .34127 .04841 m .34502 .04841 L s .34127 .06526 m .34502 .06526 L s .34127 .0821 m .34502 .0821 L s .34127 .1158 m .34502 .1158 L s .34127 .13265 m .34502 .13265 L s .34127 .14949 m .34502 .14949 L s .34127 .16634 m .34502 .16634 L s .34127 .20004 m .34502 .20004 L s .34127 .21688 m .34502 .21688 L s .34127 .23373 m .34502 .23373 L s .34127 .25058 m .34502 .25058 L s .34127 .28427 m .34502 .28427 L s .34127 .30112 m .34502 .30112 L s .34127 .31797 m .34502 .31797 L s .34127 .33482 m .34502 .33482 L s .34127 .36851 m .34502 .36851 L s .34127 .38536 m .34502 .38536 L s .34127 .40221 m .34502 .40221 L s .34127 .41905 m .34502 .41905 L s .34127 .45275 m .34502 .45275 L s .34127 .4696 m .34502 .4696 L s .34127 .48644 m .34502 .48644 L s .34127 .50329 m .34502 .50329 L s .34127 .53699 m .34502 .53699 L s .34127 .55383 m .34502 .55383 L s .34127 .57068 m .34502 .57068 L s .34127 .58753 m .34502 .58753 L s .25 Mabswid .34127 0 m .34127 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath MathSubEnd P % End of sub-graphic % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{185, 114.188}, ImageMargins->{{44, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg@3oool010000000oooo0?ooo`0000030?ooo`030000003oool0oooo07H0oooo 000k0?ooo`030000003oool0oooo0080oooo0P00001_0?ooo`030000003oool0oooo00D0oooo000c 0?ooo`@000001@3oool01@000000oooo0?ooo`3oool0000006l0oooo00<000000?ooo`3oool01P3o ool003T0oooo00@000000?ooo`3oool000000`3oool00`000000oooo0?ooo`1/0?ooo`0300000000 ool0oooo00L0oooo000j0?ooo`800000103oool2000006`0oooo00<00000003oo`00ool0203oool0 0400oooo00<000000?ooo`3oool0JP3oool0100000000?oo003oo`00ool80?ooo`00@03oool00`00 0000oooo0?ooo`1Y0?ooo`0300000000ool00?oo00800?oo203oool00400oooo0P00001Y0?ooo`03 00000000ool00?oo00<00?oo203oool00400oooo00<000000?ooo`3oool0I`3oool00`0000000?oo 003oo`04003oo`P0oooo00100?ooo`030000003oool0oooo06H0oooo00<00000003oo`00ool01@00 ool80?ooo`00@03oool2000006H0oooo00<00000003oo`00ool01P00ool80?ooo`00@03oool00`00 0000oooo0?ooo`1S0?ooo`8000002@00ool80?ooo`00@03oool00`000000oooo0?ooo`1R0?ooo`03 00000000ool00?oo00T00?oo203oool00400oooo0P00001R0?ooo`0300000000ool00?oo00X00?oo 203oool003T0oooo100000030?ooo`030000003oool0oooo0600oooo00<00000003oo`00ool02`00 ool80?ooo`00>@3oool00`000000oooo0?ooo`040?ooo`030000003oool0oooo05l0oooo00<00000 003oo`00ool03000ool80?ooo`00>P3oool00`000000oooo0?ooo`030?ooo`800000G`3oool00`00 00000?oo003oo`0=003oo`P0oooo000c0?ooo`@00000103oool00`000000oooo0?ooo`020?ooo`03 0000003oool0oooo05`0oooo0P00000@003oo`P0oooo000i0?ooo`040000003oool0oooo000000<0 oooo00<000000?ooo`3oool0F`3oool00`0000000?oo003oo`0@003oo`P0oooo000j0?ooo`800000 103oool2000005/0oooo00<00000003oo`00ool04@00ool80?ooo`00@03oool00`000000oooo0?oo o`1I0?ooo`0300000000ool00?oo01800?oo203oool00400oooo00<000000?ooo`3oool0F03oool0 0`0000000?oo003oo`0C003oo`P0oooo00100?ooo`800000F03oool00`0000000?oo003oo`0D003o o`P0oooo00100?ooo`030000003oool0oooo05D0oooo0P00000G003oo`P0oooo00100?ooo`030000 003oool0oooo05@0oooo00<00000003oo`00ool05`00ool80?ooo`00@03oool2000005@0oooo00<0 0000003oo`00ool06000ool80?ooo`00@03oool00`000000oooo0?ooo`1B0?ooo`0300000000ool0 0?oo01T00?oo203oool00400oooo00<000000?ooo`3oool0D@3oool00`0000000?oo003oo`0J003o o`P0oooo00100?ooo`800000D@3oool00`0000000?oo003oo`0K003oo`P0oooo000i0?ooo`D00000 0P3oool00`000000oooo0?ooo`1>0?ooo`8000007P00ool80?ooo`00>`3oool00`000000oooo0?oo o`020?ooo`030000003oool0oooo04d0oooo00<00000003oo`00ool07P00ool80?ooo`009@3ooolf 000003@0oooo00<00000003oo`00ool07`00ool80?ooo`00803oool5000000h00?oo10000004003o o`0300000000ool00?oo00800?oo00<00000003oo`00ool06000ool7000002`0oooo00<00000003o o`00ool08000ool80?ooo`00703oool4000001/00?oo00<00000003oo`00ool00P00ool00`000000 0?oo003oo`0O003oo`@000009P3oool2000002<00?oo203oool001T0oooo0`00000N003oo`800000 1000ool2000002@00?oo0P00000S0?ooo`0300000000ool00?oo02<00?oo203oool000T0oooo1@00 00080?ooo`<000001`00ool2000000@00?oo0P000004003oo`8000004P00ool00`0000000?oo003o o`0B003oo`8000001000ool2000000@00?oo0P000005003oo`800000303oool5000000l0oooo00<0 0000003oo`00000010000002003oo`8000001000ool2000001@00?oo100000060?ooo`002`3oool0 0`000000oooo0?ooo`060?ooo`8000002@00ool0100000000?oo003oo`000008003oo`0400000000 ool00?oo000001400?oo00<00000003oo`00ool04@00ool0100000000?oo003oo`000008003oo`04 00000000ool00?oo000000H00?oo0`00000;0?ooo`030000003oool0oooo00d0oooo0P000004003o o`0300000000ool00?oo00L00?oo00@00000003oo`00ool000004`00ool00`0000000?oo0?ooo`07 0?ooo`002`3oool00`000000oooo0?ooo`050?ooo`0300000000ool00?oo00T00?oo00@00000003o o`00ool000002`00ool00`0000000?oo003oo`0?003oo`8000004P00ool0100000000?oo003oo`00 000;003oo`0300000000ool00?oo00L00?oo0P0000090?ooo`030000003oool0oooo00`0oooo00<0 0000003oo`00ool01000ool00`0000000?oo003oo`0:003oo`0300000000ool00?oo01800?oo00<0 00000?ooo`3oool01P3oool000<0oooo100000040?ooo`030000003oool0oooo00<0oooo0P000006 003oo`@000000P00ool0100000000?oo003oo`000009003oo`8000004P00ool00`0000000?oo003o o`0A003oo`0400000000ool00?oo000000T00?oo0P00000<003oo`8000001`3oool00`000000oooo 0?ooo`0:0?ooo`8000001`00ool00`0000000?oo003oo`08003oo`8000005P00ool00`000000oooo 0?ooo`050?ooo`002`3oool010000000oooo0?ooo`3oool2000000h00?oo00@00000003oo`00ool0 00002@00ool00`0000000?oo003oo`0A003oo`0300000000ool00?oo01400?oo00@00000003oo`00 ool000002@00ool00`0000000?oo003oo`0=003oo`8000001@3oool00`000000oooo0?ooo`080?oo o`8000002@00ool00`0000000?oo003oo`08003oo`0300000000ool00?oo01<00?oo00@00000003o o`3oool000001P3oool000X0oooo0P0000020?ooo`0300000000ool00?oo00l00?oo0P00000:003o o`<000004@00ool2000001<00?oo0P00000:003oo`<000003`00ool010000000oooo0?ooo`3oool2 000000P0oooo0P00000:003oo`8000002P00ool3000001@00?oo0P0000070?ooo`003@3oool00`00 00000?oo003oo`0`003oo`0300000000ool00?oo03400?oo00<000000?ooo`3oool02@3oool00`00 00000?oo003oo`0^003oo`P0oooo000<0?ooo`0300000000ool00?oo03400?oo0P00000c003oo`03 0000003oool0oooo00H0oooo0P00000a003oo`P0oooo000;0?ooo`0300000000ool00?oo03800?oo 00<00000003oo`00ool00?ooo`0300000000ool00?oo03400?oo203oool001D0oooo0`00000X003oo`0300000000ool0 0?oo02L00?oo0`00000A0?ooo`0300000000ool00?oo03000?oo203oool001P0oooo0P00000V003o o`0300000000ool00?oo02<00?oo1000000D0?ooo`0300000000ool00?oo03000?oo203oool001X0 oooo0P00000T003oo`8000008@00ool3000001T0oooo00<00000003oo`00ool0;`00ool80?ooo`00 703oool3000002400?oo00<00000003oo`00ool07000ool4000001d0oooo00<00000003oo`00ool0 ;P00ool80?ooo`007`3oool4000001d00?oo00<00000003oo`00ool06@00ool300000240oooo00<0 0000003oo`00ool0;P00ool80?ooo`008`3oool5000001P00?oo0P00000F003oo`@000009@3oool0 0`0000000?oo003oo`0]003oo`P0oooo000X0?ooo`D000003000ool5000000800?oo00<00000003o o`00ool04P00ool3000002T0oooo00<00000003oo`00ool0;@00ool80?ooo`00;@3oool6000000P0 0?oo00<00000003oo`00ool00P00ool00`0000000?oo003oo`0;003oo`L00000;@3oool00`000000 0?oo003oo`0/003oo`P0oooo000c0?oooa/00000=03oool00`0000000?oo003oo`0/003oo`P0oooo 000k0?ooo`030000003oool0oooo0080oooo00<000000?ooo`3oool0@03oool00`0000000?oo003o o`0[003oo`P0oooo000k0?ooo`030000003oool0oooo0080oooo00<000000?ooo`3oool0@03oool0 0`0000000?oo003oo`0[003oo`P0oooo000j0?ooo`800000103oool200000480oooo00<00000003o o`00ool0:P00ool80?ooo`00@03oool00`000000oooo0?ooo`110?ooo`0300000000ool00?oo02X0 0?oo203oool00400oooo00<000000?ooo`3oool0@P3oool00`0000000?oo003oo`0Y003oo`P0oooo 00100?ooo`800000@`3oool00`0000000?oo003oo`0Y003oo`P0oooo00100?ooo`030000003oool0 oooo04<0oooo00<00000003oo`00ool0:000ool80?ooo`00@03oool00`000000oooo0?ooo`130?oo o`0300000000ool00?oo02P00?oo203oool00400oooo0P0000150?ooo`0300000000ool00?oo02L0 0?oo203oool00400oooo00<000000?ooo`3oool0A03oool00`0000000?oo003oo`0W003oo`P0oooo 00100?ooo`030000003oool0oooo04D0oooo00<00000003oo`00ool09P00ool80?ooo`00@03oool2 000004H0oooo00<00000003oo`00ool09P00ool80?ooo`00>@3oool4000000<0oooo00<000000?oo o`3oool0AP3oool00`0000000?oo003oo`0U003oo`P0oooo000i0?ooo`030000003oool0oooo00@0 oooo00<000000?ooo`3oool0AP3oool00`0000000?oo003oo`0U003oo`P0oooo000j0?ooo`030000 003oool0oooo00<0oooo0P0000180?ooo`0300000000ool00?oo02@00?oo203oool003/0oooo00<0 00000?ooo`3oool00P3oool00`000000oooo0?ooo`170?ooo`0300000000ool00?oo02@00?oo203o ool003T0oooo00@000000?ooo`3oool000000`3oool00`000000oooo0?ooo`180?ooo`0300000000 ool00?oo02<00?oo203oool003X0oooo0P0000040?ooo`800000B@3oool00`0000000?oo003oo`0S 003oo`P0oooo00100?ooo`030000003oool0oooo04T0oooo00<00000003oo`00ool08P00ool80?oo o`00@03oool00`000000oooo0?ooo`190?ooo`0300000000ool00?oo02800?oo203oool00400oooo 0P00001;0?ooo`0300000000ool00?oo02400?oo203oool00400oooo00<000000?ooo`3oool0BP3o ool00`0000000?oo003oo`0Q003oo`P0oooo00100?ooo`030000003oool0oooo04/0oooo00<00000 003oo`00ool08000ool80?ooo`00@03oool2000004`0oooo00<00000003oo`00ool08000ool80?oo o`00@03oool00`000000oooo0?ooo`1;0?ooo`0300000000ool00?oo02000?oo203oool00400oooo 00<000000?ooo`3oool0C03oool00`0000000?oo003oo`0O003oo`P0oooo00100?ooo`800000C@3o ool00`0000000?oo003oo`0O003oo`P0oooo000j0?ooo`800000103oool00`000000oooo0?ooo`1< 0?ooo`0300000000ool00?oo01l00?oo203oool003T0oooo00@000000?ooo`3oool000000`3oool0 0`000000oooo0?ooo`1=0?ooo`0300000000ool00?oo01h00?oo203oool003/0oooo00<000000?oo o`3oool00P3oool2000004h0oooo00<00000003oo`00ool07P00ool80?ooo`00?03oool01@000000 oooo0?ooo`3oool0000004l0oooo00<00000003oo`00ool07P00ool80?ooo`00>@3oool010000000 oooo0?ooo`0000030?ooo`030000003oool0oooo04h0oooo00<00000003oo`00ool07@00ool80?oo o`00>P3oool2000000@0oooo0P00001?0?ooo`0300000000ool00?oo01d00?oo203oool00400oooo 00<000000?ooo`3oool0CP3oool00`0000000?oo003oo`0M003oo`P0oooo00100?ooo`030000003o ool0oooo04h0oooo00<00000003oo`00ool07@00ool80?ooo`00@03oool200000500oooo00<00000 003oo`00ool07000ool80?ooo`00@03oool00`000000oooo0?ooo`1?0?ooo`0300000000ool00?oo 01`00?oo203oool00400oooo00<000000?ooo`3oool0C`3oool00`0000000?oo003oo`0L003oo`P0 oooo00100?ooo`800000D@3oool00`0000000?oo003oo`0K003oo`P0oooo00100?ooo`030000003o ool0oooo0500oooo00<00000003oo`00ool06`00ool80?ooo`00@03oool00`000000oooo0?ooo`1@ 0?ooo`0300000000ool00?oo01/00?oo203oool00400oooo0P00001B0?ooo`0300000000ool00?oo 01X00?oo203oool003/0oooo0`0000020?ooo`030000003oool0oooo0540oooo00<00000003oo`00 ool06P00ool80?ooo`00?03oool01@000000oooo0?ooo`3oool0000005<0oooo00<00000003oo`00 ool06P00ool80?ooo`00>@3oool500000080oooo0P00001C0?ooo`0300000000ool00?oo01T00?oo 203oool003T0oooo00@000000?ooo`3oool000000`3oool00`000000oooo0?ooo`1B0?ooo`030000 0000ool00?oo01T00?oo203oool003X0oooo00<000000?ooo`0000000`3oool00`000000oooo0?oo o`1B0?ooo`0300000000ool00?oo01T00?oo203oool003/0oooo0P00001I0?oooa/00?oo203oool0 09H0oooo6`00ool80?ooo`00\ \>"], ImageRangeCache->{{{0, 184}, {113.188, 0}} -> {-0.00234134, 0, 0.00546023, \ 0.00883484}, {{0.375, 183.563}, {113.188, 0}} -> {-1.15557, -3.30231, \ 0.0178276, 0.0671861}}] }, Open ]], Cell[TextData[{ "\nConsider the function ", Cell[BoxData[ \(TraditionalForm\`\(\(x\^\(\(\ \)\(3\)\)\)\(-\)\(1\)\(\ \)\)\)]], "and suppose we need to find the area below its graph and above x - axis\n\ over the interval [1, 2] . " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(FilledPlot[x\^3 - 1, {x, 1, 2}, Fills \[Rule] {Hue[ .6]}];\)\(\ \)\(\[IndentingNewLine]\)\)\)], \ "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 0.618034 [ [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.928572 0.952381 0.0147151 0.0840863 [ [.21429 .00222 -9 -9 ] [.21429 .00222 9 0 ] [.40476 .00222 -9 -9 ] [.40476 .00222 9 0 ] [.59524 .00222 -9 -9 ] [.59524 .00222 9 0 ] [.78571 .00222 -9 -9 ] [.78571 .00222 9 0 ] [.97619 .00222 -3 -9 ] [.97619 .00222 3 0 ] [.01131 .0988 -6 -4.5 ] [.01131 .0988 0 4.5 ] [.01131 .18289 -6 -4.5 ] [.01131 .18289 0 4.5 ] [.01131 .26697 -6 -4.5 ] [.01131 .26697 0 4.5 ] [.01131 .35106 -6 -4.5 ] [.01131 .35106 0 4.5 ] [.01131 .43515 -6 -4.5 ] [.01131 .43515 0 4.5 ] [.01131 .51923 -6 -4.5 ] [.01131 .51923 0 4.5 ] [.01131 .60332 -6 -4.5 ] [.01131 .60332 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .01472 m .21429 .02097 L s [(1.2)] .21429 .00222 0 1 Mshowa .40476 .01472 m .40476 .02097 L s [(1.4)] .40476 .00222 0 1 Mshowa .59524 .01472 m .59524 .02097 L s [(1.6)] .59524 .00222 0 1 Mshowa .78571 .01472 m .78571 .02097 L s [(1.8)] .78571 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(2)] .97619 .00222 0 1 Mshowa .125 Mabswid .07143 .01472 m .07143 .01847 L s .11905 .01472 m .11905 .01847 L s .16667 .01472 m .16667 .01847 L s .2619 .01472 m .2619 .01847 L s .30952 .01472 m .30952 .01847 L s .35714 .01472 m .35714 .01847 L s .45238 .01472 m .45238 .01847 L s .5 .01472 m .5 .01847 L s .54762 .01472 m .54762 .01847 L s .64286 .01472 m .64286 .01847 L s .69048 .01472 m .69048 .01847 L s .7381 .01472 m .7381 .01847 L s .83333 .01472 m .83333 .01847 L s .88095 .01472 m .88095 .01847 L s .92857 .01472 m .92857 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .02381 .0988 m .03006 .0988 L s [(1)] .01131 .0988 1 0 Mshowa .02381 .18289 m .03006 .18289 L s [(2)] .01131 .18289 1 0 Mshowa .02381 .26697 m .03006 .26697 L s [(3)] .01131 .26697 1 0 Mshowa .02381 .35106 m .03006 .35106 L s [(4)] .01131 .35106 1 0 Mshowa .02381 .43515 m .03006 .43515 L s [(5)] .01131 .43515 1 0 Mshowa .02381 .51923 m .03006 .51923 L s [(6)] .01131 .51923 1 0 Mshowa .02381 .60332 m .03006 .60332 L s [(7)] .01131 .60332 1 0 Mshowa .125 Mabswid .02381 .03153 m .02756 .03153 L s .02381 .04835 m .02756 .04835 L s .02381 .06517 m .02756 .06517 L s .02381 .08198 m .02756 .08198 L s .02381 .11562 m .02756 .11562 L s .02381 .13244 m .02756 .13244 L s .02381 .14925 m .02756 .14925 L s .02381 .16607 m .02756 .16607 L s .02381 .1997 m .02756 .1997 L s .02381 .21652 m .02756 .21652 L s .02381 .23334 m .02756 .23334 L s .02381 .25016 m .02756 .25016 L s .02381 .28379 m .02756 .28379 L s .02381 .30061 m .02756 .30061 L s .02381 .31743 m .02756 .31743 L s .02381 .33424 m .02756 .33424 L s .02381 .36788 m .02756 .36788 L s .02381 .38469 m .02756 .38469 L s .02381 .40151 m .02756 .40151 L s .02381 .41833 m .02756 .41833 L s .02381 .45196 m .02756 .45196 L s .02381 .46878 m .02756 .46878 L s .02381 .4856 m .02756 .4856 L s .02381 .50242 m .02756 .50242 L s .02381 .53605 m .02756 .53605 L s .02381 .55287 m .02756 .55287 L s .02381 .56968 m .02756 .56968 L s .02381 .5865 m .02756 .5865 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .01472 m .06244 .02537 L .10458 .03797 L .14415 .05079 L .18221 .06404 L .22272 .07917 L .26171 .09478 L .30316 .11253 L .34309 .1308 L .3815 .14949 L .42237 .17062 L .46172 .19221 L .49955 .21415 L .53984 .23883 L .57861 .26389 L .61984 .292 L .65954 .32052 L .69774 .34933 L .73838 .38151 L .77751 .41401 L .81909 .45023 L .85916 .48679 L .89771 .52355 L .93871 .56439 L .97619 .60332 L s 0 .4 1 r .02381 .01472 m .06244 .02537 L .10458 .03797 L .14415 .05079 L .18221 .06404 L .22272 .07917 L .26171 .09478 L .30316 .11253 L .34309 .1308 L .3815 .14949 L .42237 .17062 L .46172 .19221 L .49955 .21415 L .53984 .23883 L .57861 .26389 L .61984 .292 L .65954 .32052 L .69774 .34933 L .73838 .38151 L .77751 .41401 L .81909 .45023 L .85916 .48679 L .89771 .52355 L .93871 .56439 L .97619 .60332 L .97619 .01472 L .02381 .01472 L F 0 g .02381 .01472 m .06244 .02537 L .10458 .03797 L .14415 .05079 L .18221 .06404 L .22272 .07917 L .26171 .09478 L .30316 .11253 L .34309 .1308 L .3815 .14949 L .42237 .17062 L .46172 .19221 L .49955 .21415 L .53984 .23883 L .57861 .26389 L .61984 .292 L .65954 .32052 L .69774 .34933 L .73838 .38151 L .77751 .41401 L .81909 .45023 L .85916 .48679 L .89771 .52355 L .93871 .56439 L .97619 .60332 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.928572 0.952381 0.0147151 0.0840863 [ [.21429 .00222 -9 -9 ] [.21429 .00222 9 0 ] [.40476 .00222 -9 -9 ] [.40476 .00222 9 0 ] [.59524 .00222 -9 -9 ] [.59524 .00222 9 0 ] [.78571 .00222 -9 -9 ] [.78571 .00222 9 0 ] [.97619 .00222 -3 -9 ] [.97619 .00222 3 0 ] [.01131 .0988 -6 -4.5 ] [.01131 .0988 0 4.5 ] [.01131 .18289 -6 -4.5 ] [.01131 .18289 0 4.5 ] [.01131 .26697 -6 -4.5 ] [.01131 .26697 0 4.5 ] [.01131 .35106 -6 -4.5 ] [.01131 .35106 0 4.5 ] [.01131 .43515 -6 -4.5 ] [.01131 .43515 0 4.5 ] [.01131 .51923 -6 -4.5 ] [.01131 .51923 0 4.5 ] [.01131 .60332 -6 -4.5 ] [.01131 .60332 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .01472 m .21429 .02097 L s [(1.2)] .21429 .00222 0 1 Mshowa .40476 .01472 m .40476 .02097 L s [(1.4)] .40476 .00222 0 1 Mshowa .59524 .01472 m .59524 .02097 L s [(1.6)] .59524 .00222 0 1 Mshowa .78571 .01472 m .78571 .02097 L s [(1.8)] .78571 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(2)] .97619 .00222 0 1 Mshowa .125 Mabswid .07143 .01472 m .07143 .01847 L s .11905 .01472 m .11905 .01847 L s .16667 .01472 m .16667 .01847 L s .2619 .01472 m .2619 .01847 L s .30952 .01472 m .30952 .01847 L s .35714 .01472 m .35714 .01847 L s .45238 .01472 m .45238 .01847 L s .5 .01472 m .5 .01847 L s .54762 .01472 m .54762 .01847 L s .64286 .01472 m .64286 .01847 L s .69048 .01472 m .69048 .01847 L s .7381 .01472 m .7381 .01847 L s .83333 .01472 m .83333 .01847 L s .88095 .01472 m .88095 .01847 L s .92857 .01472 m .92857 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .02381 .0988 m .03006 .0988 L s [(1)] .01131 .0988 1 0 Mshowa .02381 .18289 m .03006 .18289 L s [(2)] .01131 .18289 1 0 Mshowa .02381 .26697 m .03006 .26697 L s [(3)] .01131 .26697 1 0 Mshowa .02381 .35106 m .03006 .35106 L s [(4)] .01131 .35106 1 0 Mshowa .02381 .43515 m .03006 .43515 L s [(5)] .01131 .43515 1 0 Mshowa .02381 .51923 m .03006 .51923 L s [(6)] .01131 .51923 1 0 Mshowa .02381 .60332 m .03006 .60332 L s [(7)] .01131 .60332 1 0 Mshowa .125 Mabswid .02381 .03153 m .02756 .03153 L s .02381 .04835 m .02756 .04835 L s .02381 .06517 m .02756 .06517 L s .02381 .08198 m .02756 .08198 L s .02381 .11562 m .02756 .11562 L s .02381 .13244 m .02756 .13244 L s .02381 .14925 m .02756 .14925 L s .02381 .16607 m .02756 .16607 L s .02381 .1997 m .02756 .1997 L s .02381 .21652 m .02756 .21652 L s .02381 .23334 m .02756 .23334 L s .02381 .25016 m .02756 .25016 L s .02381 .28379 m .02756 .28379 L s .02381 .30061 m .02756 .30061 L s .02381 .31743 m .02756 .31743 L s .02381 .33424 m .02756 .33424 L s .02381 .36788 m .02756 .36788 L s .02381 .38469 m .02756 .38469 L s .02381 .40151 m .02756 .40151 L s .02381 .41833 m .02756 .41833 L s .02381 .45196 m .02756 .45196 L s .02381 .46878 m .02756 .46878 L s .02381 .4856 m .02756 .4856 L s .02381 .50242 m .02756 .50242 L s .02381 .53605 m .02756 .53605 L s .02381 .55287 m .02756 .55287 L s .02381 .56968 m .02756 .56968 L s .02381 .5865 m .02756 .5865 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath MathSubEnd P % End of sub-graphic % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{186.875, 115.438}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg0?ooo`030000003oool0oooo00X0oooo0`00002@001Vo``0oooo000>0?ooo`030000 003oool0oooo00d0oooo1000002<001Vo``0oooo000>0?ooo`8000004P3oool3000008T006Ko303o ool000h0oooo00<000000?ooo`3oool0503oool3000008H006Ko303oool000h0oooo00<000000?oo o`3oool05`3oool3000008<006Ko303oool000h0oooo0P00000K0?ooo`800000P@00I_l<0?ooo`00 3P3oool00`000000oooo0?ooo`0L0?ooo`800000O`00I_l<0?ooo`003P3oool2000001l0oooo0P00 001m001Vo``0oooo00070?ooo`D000000P3oool00`000000oooo0?ooo`0P0?ooo`800000N`00I_l< 0?ooo`002@3oool00`000000oooo0?ooo`020?ooo`030000003oool0oooo0280oooo0P00001i001V o``0oooo00090?ooo`030000003oool0oooo0080oooo0`00000T0?ooo`<00000MP00I_l<0?ooo`00 2@3oool00`000000oooo0?ooo`020?ooo`030000003oool0oooo02L0oooo0P00001d001Vo``0oooo 00090?ooo`030000003oool0oooo0080oooo00<000000?ooo`3oool0:@3oool20000078006Ko303o ool000P0oooo0P0000040?ooo`800000;03oool3000006l006Ko303oool000h0oooo00<000000?oo o`3oool0;P3oool2000006d006Ko303oool000h0oooo00<000000?ooo`3oool0<03oool2000006/0 06Ko303oool000h0oooo0P00000c0?ooo`<00000J000I_l<0?ooo`003P3oool00`000000oooo0?oo o`0e0?ooo`800000IP00I_l<0?ooo`003P3oool00`000000oooo0?ooo`0g0?ooo`800000I000I_l< 0?ooo`003P3oool2000003X0oooo0P00001R001Vo``0oooo000>0?ooo`030000003oool0oooo03/0 oooo0P00001P001Vo``0oooo000>0?ooo`030000003oool0oooo03d0oooo0P00001N001Vo``0oooo 00070?ooo`@000000`3oool200000400oooo0P00001L001Vo``0oooo00070?ooo`030000003oool0 oooo00@0oooo00<000000?ooo`3oool0@@3oool2000005X006Ko303oool000P0oooo00<000000?oo o`3oool00`3oool3000004<0oooo0P00001H001Vo``0oooo00090?ooo`030000003oool0oooo0080 oooo00<000000?ooo`3oool0A@3oool2000005H006Ko303oool000L0oooo00@000000?ooo`3oool0 00000`3oool00`000000oooo0?ooo`170?ooo`0300000000I_l006Ko05<006Ko303oool000P0oooo 0P0000040?ooo`800000B@3oool2000005<006Ko303oool000h0oooo00<000000?ooo`3oool0BP3o ool00`00000006Ko001Vo`1@001Vo``0oooo000>0?ooo`030000003oool0oooo04/0oooo0P00001@ 001Vo``0oooo000>0?ooo`800000CP3oool00`00000006Ko001Vo`1=001Vo``0oooo000>0?ooo`03 0000003oool0oooo04h0oooo0P00001=001Vo``0oooo000>0?ooo`030000003oool0oooo0500oooo 0P00001;001Vo``0oooo000>0?ooo`800000D`3oool2000004T006Ko303oool000h0oooo00<00000 0?ooo`3oool0E03oool00`00000006Ko001Vo`16001Vo``0oooo000>0?ooo`030000003oool0oooo 05D0oooo0P000016001Vo``0oooo000>0?ooo`800000F03oool2000004@006Ko303oool000P0oooo 0P0000040?ooo`030000003oool0oooo05T0oooo0P000012001Vo``0oooo00070?ooo`040000003o ool0oooo000000<0oooo00<000000?ooo`3oool0F`3oool00`00000006Ko001Vo`0o001Vo``0oooo 00090?ooo`030000003oool0oooo0080oooo0`00001L0?ooo`800000?`00I_l<0?ooo`002P3oool0 1@000000oooo0?ooo`3oool000000600oooo00<00000001Vo`00I_l0?000I_l<0?ooo`001`3oool0 10000000oooo0?ooo`0000030?ooo`800000H03oool00`00000006Ko001Vo`0k001Vo``0oooo0008 0?ooo`800000103oool00`000000oooo0?ooo`1P0?ooo`800000>`00I_l<0?ooo`003P3oool00`00 0000oooo0?ooo`1R0?ooo`0300000000I_l006Ko03P006Ko303oool000h0oooo0P00001T0?ooo`03 00000000I_l006Ko03L006Ko303oool000h0oooo00<000000?ooo`3oool0I03oool00`00000006Ko 001Vo`0f001Vo``0oooo000>0?ooo`030000003oool0oooo06D0oooo0P00000f001Vo``0oooo000> 0?ooo`800000J03oool00`00000006Ko001Vo`0c001Vo``0oooo000>0?ooo`030000003oool0oooo 06P0oooo00<00000001Vo`00I_l00?ooo`030000 003oool0oooo07@0oooo00<00000001Vo`00I_l09P00I_l<0?ooo`003P3oool2000007H0oooo00<0 0000001Vo`00I_l09@00I_l<0?ooo`003P3oool00`000000oooo0?ooo`1f0?ooo`8000009@00I_l< 0?ooo`003P3oool00`000000oooo0?ooo`1h0?ooo`0300000000I_l006Ko028006Ko303oool000h0 oooo0P00001j0?ooo`0300000000I_l006Ko024006Ko303oool000h0oooo00<000000?ooo`3oool0 NP3oool20000024006Ko303oool000h0oooo00<000000?ooo`3oool0O03oool00`00000006Ko001V o`0N001Vo``0oooo000>0?ooo`800000OP3oool00`00000006Ko001Vo`0M001Vo``0oooo00080?oo o`800000103oool00`000000oooo0?ooo`1n0?ooo`0300000000I_l006Ko01`006Ko303oool000L0 oooo00@000000?ooo`3oool000000`3oool00`000000oooo0?ooo`1o0?ooo`8000007000I_l<0?oo o`002P3oool010000000oooo0?ooo`3oool300000840oooo00<00000001Vo`00I_l06@00I_l<0?oo o`00203oool2000000@0oooo00<000000?ooo`3oool0PP3oool00`00000006Ko001Vo`0H001Vo``0 oooo00080?ooo`030000003oool0oooo00<0oooo00<000000?ooo`3oool0P`3oool00`00000006Ko 001Vo`0G001Vo``0oooo00080?ooo`<000000`3oool2000008D0oooo00<00000001Vo`00I_l05P00 I_l<0?ooo`003P3oool00`000000oooo0?ooo`250?ooo`0300000000I_l006Ko01D006Ko303oool0 00h0oooo0P0000270?ooo`8000005@00I_l<0?ooo`003P3oool00`000000oooo0?ooo`280?ooo`03 00000000I_l006Ko018006Ko303oool000h0oooo00<000000?ooo`3oool0R@3oool00`00000006Ko 001Vo`0A001Vo``0oooo000>0?ooo`800000R`3oool00`00000006Ko001Vo`0@001Vo``0oooo000> 0?ooo`030000003oool0oooo08/0oooo00<00000001Vo`00I_l03`00I_l<0?ooo`003P3oool00`00 0000oooo0?ooo`2<0?ooo`0300000000I_l006Ko00h006Ko303oool000h0oooo0P00002>0?ooo`03 00000000I_l006Ko00d006Ko303oool000L0oooo0`0000040?ooo`030000003oool0oooo08d0oooo 00<00000001Vo`00I_l03@00I_l<0?ooo`001`3oool010000000oooo0?ooo`0000030?ooo`030000 003oool0oooo08h0oooo00<00000001Vo`00I_l03000I_l<0?ooo`001`3oool010000000oooo0?oo o`0000030?ooo`<00000S`3oool00`00000006Ko001Vo`0;001Vo``0oooo00070?ooo`<00000103o ool00`000000oooo0?ooo`2@0?ooo`0300000000I_l006Ko00X006Ko303oool000P0oooo00<00000 0?ooo`3oool00`3oool00`000000oooo0?ooo`2A0?ooo`0300000000I_l006Ko00T006Ko303oool0 00P0oooo0`0000030?ooo`800000T`3oool00`00000006Ko001Vo`08001Vo``0oooo000>0?ooo`03 0000003oool0oooo09<0oooo0P000008001Vo``0oooo000>0?ooo`030000003oool0oooo09D0oooo 00<00000001Vo`00I_l01@00I_l<0?ooo`003P3oool2000009L0oooo00<00000001Vo`00I_l01000 I_l<0?ooo`003P3oool00`000000oooo0?ooo`2G0?ooo`0300000000I_l006Ko00<006Ko303oool0 00h0oooo0P00002I0?ooo`0300000000I_l006Ko008006Ko303oool000h0oooo00<000000?ooo`3o ool0V@3oool01000000006Ko001Vo`00I_l<0?ooo`003P3oool00`000000oooo0?ooo`2J0?ooo`03 00000000I_l006Ko00`0oooo000>0?ooo`800000V`3oool00`00000006Ko001Vo`0<0?ooo`002@3o ool00`000000oooo0?ooo`020?ooo`030000003oool0oooo09/0oooo00<00000001Vo`3oool02`3o ool000T0oooo00<000000?ooo`3oool00P3oool00`000000oooo0?ooo`2L0?ooo`030000003oool0 oooo00X0oooo00090?ooo`030000003oool0oooo0080oooo0`00002M0?ooo`030000003oool0oooo 00T0oooo000:0?ooo`050000003oool0oooo0?ooo`000000Z`3oool000L0oooo00@000000?ooo`3o ool000000`3oool00`000000oooo0?ooo`2Y0?ooo`001`3oool400000:l0oooo002j0?ooo`00\ \>"], ImageRangeCache->{{{0, 185.875}, {114.438, 0}} -> {-0.00192234, \ -5.00002*^-06, 0.00540064, 0.00873843}, {{0.3125, 185.5}, {114.438, 0}} -> \ {0.906237, -0.78604, 0.00625299, 0.0708228}}] }, Open ]], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\(<< Graphics`Graphics`\)\(\[IndentingNewLine]\)\( (*\ loads\ DisplayTogether[ . ], GeneralizedBarChar[ . ]\ *) \)\)\)\)], "Input"], Cell[BoxData[ \(f[x_] := x\^3 - 1; \ a = 1; \ b = 2; \ n = 20; \ dx = \(b - a\)\/n;\)], "Input"], Cell[BoxData[ \(RiemannRight[f_, a_, b_, n_] := \[IndentingNewLine]\[Sum]\+\(i = 1\)\%n\(\( f[ a + i*\(b - a\)\/n]\)\(*\)\(\(b - a\)\/n\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\( \ (*\ right\ end\ point\ Riemann\ sum\ *) \)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(DisplayTogether[ GeneralizedBarChart[\[IndentingNewLine]Table[{a + dx/2 + \((i - 1)\)*dx, f[a + i*\(b - a\)\/n], dx}, {i, 1, n}]], Plot[f[x], {x, a, b}]];\)\(\n\)\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.928571 0.952381 0.0147151 0.0840863 [ [.21429 .00222 -9 -9 ] [.21429 .00222 9 0 ] [.40476 .00222 -9 -9 ] [.40476 .00222 9 0 ] [.59524 .00222 -9 -9 ] [.59524 .00222 9 0 ] [.78571 .00222 -9 -9 ] [.78571 .00222 9 0 ] [.97619 .00222 -3 -9 ] [.97619 .00222 3 0 ] [.01131 .0988 -6 -4.5 ] [.01131 .0988 0 4.5 ] [.01131 .18289 -6 -4.5 ] [.01131 .18289 0 4.5 ] [.01131 .26697 -6 -4.5 ] [.01131 .26697 0 4.5 ] [.01131 .35106 -6 -4.5 ] [.01131 .35106 0 4.5 ] [.01131 .43515 -6 -4.5 ] [.01131 .43515 0 4.5 ] [.01131 .51923 -6 -4.5 ] [.01131 .51923 0 4.5 ] [.01131 .60332 -6 -4.5 ] [.01131 .60332 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .01472 m .21429 .02097 L s [(1.2)] .21429 .00222 0 1 Mshowa .40476 .01472 m .40476 .02097 L s [(1.4)] .40476 .00222 0 1 Mshowa .59524 .01472 m .59524 .02097 L s [(1.6)] .59524 .00222 0 1 Mshowa .78571 .01472 m .78571 .02097 L s [(1.8)] .78571 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(2)] .97619 .00222 0 1 Mshowa .125 Mabswid .07143 .01472 m .07143 .01847 L s .11905 .01472 m .11905 .01847 L s .16667 .01472 m .16667 .01847 L s .2619 .01472 m .2619 .01847 L s .30952 .01472 m .30952 .01847 L s .35714 .01472 m .35714 .01847 L s .45238 .01472 m .45238 .01847 L s .5 .01472 m .5 .01847 L s .54762 .01472 m .54762 .01847 L s .64286 .01472 m .64286 .01847 L s .69048 .01472 m .69048 .01847 L s .7381 .01472 m .7381 .01847 L s .83333 .01472 m .83333 .01847 L s .88095 .01472 m .88095 .01847 L s .92857 .01472 m .92857 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .02381 .0988 m .03006 .0988 L s [(1)] .01131 .0988 1 0 Mshowa .02381 .18289 m .03006 .18289 L s [(2)] .01131 .18289 1 0 Mshowa .02381 .26697 m .03006 .26697 L s [(3)] .01131 .26697 1 0 Mshowa .02381 .35106 m .03006 .35106 L s [(4)] .01131 .35106 1 0 Mshowa .02381 .43515 m .03006 .43515 L s [(5)] .01131 .43515 1 0 Mshowa .02381 .51923 m .03006 .51923 L s [(6)] .01131 .51923 1 0 Mshowa .02381 .60332 m .03006 .60332 L s [(7)] .01131 .60332 1 0 Mshowa .125 Mabswid .02381 .03153 m .02756 .03153 L s .02381 .04835 m .02756 .04835 L s .02381 .06517 m .02756 .06517 L s .02381 .08198 m .02756 .08198 L s .02381 .11562 m .02756 .11562 L s .02381 .13244 m .02756 .13244 L s .02381 .14925 m .02756 .14925 L s .02381 .16607 m .02756 .16607 L s .02381 .1997 m .02756 .1997 L s .02381 .21652 m .02756 .21652 L s .02381 .23334 m .02756 .23334 L s .02381 .25016 m .02756 .25016 L s .02381 .28379 m .02756 .28379 L s .02381 .30061 m .02756 .30061 L s .02381 .31743 m .02756 .31743 L s .02381 .33424 m .02756 .33424 L s .02381 .36788 m .02756 .36788 L s .02381 .38469 m .02756 .38469 L s .02381 .40151 m .02756 .40151 L s .02381 .41833 m .02756 .41833 L s .02381 .45196 m .02756 .45196 L s .02381 .46878 m .02756 .46878 L s .02381 .4856 m .02756 .4856 L s .02381 .50242 m .02756 .50242 L s .02381 .53605 m .02756 .53605 L s .02381 .55287 m .02756 .55287 L s .02381 .56968 m .02756 .56968 L s .02381 .5865 m .02756 .5865 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 1 0 0 r .02381 .01472 m .02381 .02797 L .07143 .02797 L .07143 .01472 L F 0 g .5 Mabswid .02381 .01472 m .02381 .02797 L .07143 .02797 L .07143 .01472 L .02381 .01472 L s 1 0 0 r .07143 .01472 m .07143 .04255 L .11905 .04255 L .11905 .01472 L F 0 g .07143 .01472 m .07143 .04255 L .11905 .04255 L .11905 .01472 L .07143 .01472 L s 1 0 0 r .11905 .01472 m .11905 .05851 L .16667 .05851 L .16667 .01472 L F 0 g .11905 .01472 m .11905 .05851 L .16667 .05851 L .16667 .01472 L .11905 .01472 L s 1 0 0 r .16667 .01472 m .16667 .07593 L .21429 .07593 L .21429 .01472 L F 0 g .16667 .01472 m .16667 .07593 L .21429 .07593 L .21429 .01472 L .16667 .01472 L s 1 0 0 r .21429 .01472 m .21429 .09486 L .2619 .09486 L .2619 .01472 L F 0 g .21429 .01472 m .21429 .09486 L .2619 .09486 L .2619 .01472 L .21429 .01472 L s 1 0 0 r .2619 .01472 m .2619 .11537 L .30952 .11537 L .30952 .01472 L F 0 g .2619 .01472 m .2619 .11537 L .30952 .11537 L .30952 .01472 L .2619 .01472 L s 1 0 0 r .30952 .01472 m .30952 .13751 L .35714 .13751 L .35714 .01472 L F 0 g .30952 .01472 m .30952 .13751 L .35714 .13751 L .35714 .01472 L .30952 .01472 L s 1 0 0 r .35714 .01472 m .35714 .16136 L .40476 .16136 L .40476 .01472 L F 0 g .35714 .01472 m .35714 .16136 L .40476 .16136 L .40476 .01472 L .35714 .01472 L s 1 0 0 r .40476 .01472 m .40476 .18698 L .45238 .18698 L .45238 .01472 L F 0 g .40476 .01472 m .40476 .18698 L .45238 .18698 L .45238 .01472 L .40476 .01472 L s 1 0 0 r .45238 .01472 m .45238 .21442 L .5 .21442 L .5 .01472 L F 0 g .45238 .01472 m .45238 .21442 L .5 .21442 L .5 .01472 L .45238 .01472 L s 1 0 0 r .5 .01472 m .5 .24376 L .54762 .24376 L .54762 .01472 L F 0 g .5 .01472 m .5 .24376 L .54762 .24376 L .54762 .01472 L .5 .01472 L s 1 0 0 r .54762 .01472 m .54762 .27505 L .59524 .27505 L .59524 .01472 L F 0 g .54762 .01472 m .54762 .27505 L .59524 .27505 L .59524 .01472 L .54762 .01472 L s 1 0 0 r .59524 .01472 m .59524 .30835 L .64286 .30835 L .64286 .01472 L F 0 g .59524 .01472 m .59524 .30835 L .64286 .30835 L .64286 .01472 L .59524 .01472 L s 1 0 0 r .64286 .01472 m .64286 .34374 L .69048 .34374 L .69048 .01472 L F 0 g .64286 .01472 m .64286 .34374 L .69048 .34374 L .69048 .01472 L .64286 .01472 L s 1 0 0 r .69048 .01472 m .69048 .38128 L .7381 .38128 L .7381 .01472 L F 0 g .69048 .01472 m .69048 .38128 L .7381 .38128 L .7381 .01472 L .69048 .01472 L s 1 0 0 r .7381 .01472 m .7381 .42102 L .78571 .42102 L .78571 .01472 L F 0 g .7381 .01472 m .7381 .42102 L .78571 .42102 L .78571 .01472 L .7381 .01472 L s 1 0 0 r .78571 .01472 m .78571 .46303 L .83333 .46303 L .83333 .01472 L F 0 g .78571 .01472 m .78571 .46303 L .83333 .46303 L .83333 .01472 L .78571 .01472 L s 1 0 0 r .83333 .01472 m .83333 .50738 L .88095 .50738 L .88095 .01472 L F 0 g .83333 .01472 m .83333 .50738 L .88095 .50738 L .88095 .01472 L .83333 .01472 L s 1 0 0 r .88095 .01472 m .88095 .55412 L .92857 .55412 L .92857 .01472 L F 0 g .88095 .01472 m .88095 .55412 L .92857 .55412 L .92857 .01472 L .88095 .01472 L s 1 0 0 r .92857 .01472 m .92857 .60332 L .97619 .60332 L .97619 .01472 L F 0 g .92857 .01472 m .92857 .60332 L .97619 .60332 L .97619 .01472 L .92857 .01472 L s .02381 .01472 m .06244 .02537 L .10458 .03797 L .14415 .05079 L .18221 .06404 L .22272 .07917 L .26171 .09478 L .30316 .11253 L .34309 .1308 L .3815 .14949 L .42237 .17062 L .46172 .19221 L .49955 .21415 L .53984 .23883 L .57861 .26389 L .61984 .292 L .65954 .32052 L .69774 .34933 L .73838 .38151 L .77751 .41401 L .81909 .45023 L .85916 .48679 L .89771 .52355 L .93871 .56439 L .97619 .60332 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{198, 122.313}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg0?ooo`030000003o0000o`0000<000000P3o00000`000000o`000?l00006 0?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l00005 0?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l00006 0?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l00006 0?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l00005 0?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l00006 0?l000030000003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool000l0oooo00<000000?oo o`3oool03P3oool:000000L0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l0 00030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l0 00030000003o0000o`0000H0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l0 00030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l0 00030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l0 00030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000oooo0?ooo`0:0?oo o`003`3oool00`000000oooo0?ooo`0F0?ooo`030000003o0000000000800000103o00000`000000 o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000 o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000 o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000 o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000 o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000 o`000?l000050?l000030000003oool0oooo00X0oooo000?0?ooo`030000003oool0oooo01H0oooo 00<000000?l0003o00000P3o000200000080o`0000<000000?l0003o00001@3o00000`000000o`00 0?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`00 0?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001P3o00000`000000o`00 0?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`00 0?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`00 0?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000oooo 0?ooo`0:0?ooo`003`3oool00`000000oooo0?ooo`0F0?ooo`X000001`3o00000`000000o`000?l0 00060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l0 00050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001P3o00000`000000o`000?l0 00050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l0 00060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l0 00050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000oooo0?oo o`0:0?ooo`003`3oool00`000000oooo0?ooo`0O0?ooo`<000001@3o00000`000000o`000?l00006 0?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l00005 0?l000030000003o0000o`0000H0o`0000<000000?l0003o00001P3o00000`000000o`000?l00005 0?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l00006 0?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l00005 0?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000oooo0?ooo`0: 0?ooo`003`3oool00`000000oooo0?ooo`0O0?ooo`030000003o0000o`00008000000`3o00000`00 0000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`00 0000o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001P3o00000`00 0000o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`00 0000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`00 0000o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`00 0000oooo0?ooo`0:0?ooo`001`3oool5000000<0oooo00<000000?ooo`3oool07`3oool00`000000 o`000?l000020?l00080000000<0o`000000003o00001`3o00000`000000o`000?l000050?l00003 0000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l00003 0000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l00003 0000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l00003 0000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l00003 0000003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool000T0oooo00<000000?ooo`3oool0 0`3oool00`000000oooo0?ooo`0O0?ooo`X000001`3o00000`000000o`000?l000050?l000030000 003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000 003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000 003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000 003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000 003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool000T0oooo00<000000?ooo`3oool00`3o ool2000002P0oooo00@000000?l00000000000001@3o00000`000000o`000?l000050?l000030000 003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000 003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000 003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000 003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000 003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool000T0oooo00<000000?ooo`3oool00`3o ool00`000000oooo0?ooo`0W0?ooo`040000003o0000o`000?l000<000000P3o00000`000000o`00 0?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`00 0?l000060?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`00 0?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`00 0?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`00 0?l000060?l000030000003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool000T0oooo00<0 00000?ooo`3oool00`3oool00`000000oooo0?ooo`0W0?ooo`X000001`3o00000`000000o`000?l0 00060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l0 00060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l0 00050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l0 00060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l0 00050?l000030000003oool0oooo00X0oooo00080?ooo`8000001@3oool00`000000oooo0?ooo`0` 0?ooo`<000001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<00000 0?l0003o00001P3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<00000 0?l0003o00001P3o00000`000000o`000?l000050?l000030000003o0000o`0000H0o`0000<00000 0?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<00000 0?l0003o00001P3o00000`000000o`000?l000050?l000030000003oool0oooo00X0oooo000?0?oo o`030000003oool0oooo0300oooo00<000000?l0003o00000P0000030?l000030000003o0000o`00 00H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`00 00H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`00 00D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003o0000o`00 00H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`00 00D0o`0000<000000?ooo`3oool02P3oool000l0oooo00<000000?ooo`3oool0<03oool00`000000 o`000?l000020?l00080000000<0o`000000003o00001`3o00000`000000o`000?l000050?l00003 0000003o0000o`0000H0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l00003 0000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l00003 0000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l00003 0000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000oooo0?ooo`0:0?ooo`00 3`3oool00`000000oooo0?ooo`0`0?ooo`T00000203o00000`000000o`000?l000050?l000030000 003o0000o`0000H0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000 003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000 003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000 003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000oooo0?ooo`0:0?ooo`003`3o ool00`000000oooo0?ooo`0h0?ooo`<000001P3o00000`000000o`000?l000050?l000030000003o 0000o`0000H0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003o 0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o 0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003o 0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000oooo0?ooo`0:0?ooo`003`3oool0 0`000000oooo0?ooo`0h0?ooo`030000003o0000o`0000800000103o00000`000000o`000?l00005 0?l000030000003o0000o`0000H0o`0000<000000?l0003o00001P3o00000`000000o`000?l00005 0?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l00006 0?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l00005 0?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000oooo0?ooo`0: 0?ooo`003`3oool00`000000oooo0?ooo`0h0?ooo`030000003o0000o`000080o`000P0000020?l0 00030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000060?l0 00030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l0 00030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l0 00030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l0 00030000003oool0oooo00X0oooo000?0?ooo`030000003oool0oooo03P0oooo00<000000?l0003o 0000103o0003000000L0o`0000<000000?l0003o00001P3o00000`000000o`000?l000060?l00003 0000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l00003 0000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l00003 0000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l00003 0000003oool0oooo00X0oooo000?0?ooo`030000003oool0oooo03P0oooo2`0000060?l000030000 003o0000o`0000H0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000 003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000 003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000 003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000oooo0?ooo`0:0?ooo`003`3o ool00`000000oooo0?ooo`110?ooo`040000003o00000000000000@0o`0000<000000?l0003o0000 1P3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o0000 1P3o00000`000000o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o0000 1@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o0000 1P3o00000`000000o`000?l000050?l000030000003oool0oooo00X0oooo00070?ooo`@00000103o ool00`000000oooo0?ooo`110?ooo`040000003o0000o`000?l0008000000P3o00000`000000o`00 0?l000060?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`00 0?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`00 0?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`00 0?l000060?l000030000003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool000L0oooo00<0 00000?ooo`3oool01@3oool00`000000oooo0?ooo`110?ooo`030000003o0000o`0000<0o`000`00 00080?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l0 00060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l0 00050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l0 00060?l000030000003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool000P0oooo00<00000 0?ooo`3oool0103oool200000480oooo2P0000070?l000030000003o0000o`0000H0o`0000<00000 0?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<00000 0?l0003o00001P3o00000`000000o`000?l000050?l000030000003o0000o`0000H0o`0000<00000 0?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<00000 0?ooo`3oool02P3oool000T0oooo00<000000?ooo`3oool00`3oool00`000000oooo0?ooo`190?oo o`030000003o0000000000H0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l0 00030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l0 00030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l0 00030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000oooo0?ooo`0:0?oo o`001`3oool010000000oooo0?ooo`0000040?ooo`030000003oool0oooo04T0oooo00<000000?l0 003o00000P0000040?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`00 0000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`00 0000o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`00 0000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool000P0 oooo0P0000050?ooo`030000003oool0oooo04T0oooo00<000000?l0003o00000P3o000200000080 o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003o0000o`0000H0 o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0 o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003o0000o`0000H0 o`0000<000000?l0003o00001@3o00000`000000oooo0?ooo`0:0?ooo`003`3oool00`000000oooo 0?ooo`190?ooo`030000003o0000o`0000@0o`000`0000080?l000030000003o0000o`0000D0o`00 00<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003o0000o`0000H0o`00 00<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`00 00<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003oool0oooo00X0oooo 000?0?ooo`030000003oool0oooo04T0oooo2P0000080?l000030000003o0000o`0000D0o`0000<0 00000?l0003o00001P3o00000`000000o`000?l000050?l000030000003o0000o`0000H0o`0000<0 00000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<0 00000?l0003o00001P3o00000`000000o`000?l000050?l000030000003oool0oooo00X0oooo000? 0?ooo`030000003oool0oooo0580oooo0`0000060?l000030000003o0000o`0000D0o`0000<00000 0?l0003o00001P3o00000`000000o`000?l000050?l000030000003o0000o`0000H0o`0000<00000 0?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<00000 0?l0003o00001P3o00000`000000o`000?l000050?l000030000003oool0oooo00X0oooo000?0?oo o`030000003oool0oooo0580oooo00@000000?l0003o000000001@3o00000`000000o`000?l00005 0?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l00006 0?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l00005 0?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000oooo0?ooo`0: 0?ooo`003`3oool00`000000oooo0?ooo`1B0?ooo`050000003o0000o`000?l000000000103o0000 0`000000o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o0000 0`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o0000 0`000000o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o0000 0`000000oooo0?ooo`0:0?ooo`003`3oool00`000000oooo0?ooo`1B0?ooo`030000003o0000o`00 0080o`000P0000020?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`00 0000o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`00 0000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`00 0000o`000?l000050?l000030000003oool0oooo00X0oooo000?0?ooo`030000003oool0oooo0580 oooo00<000000?l0003o0000103o00000`000000o`00000000070?l000030000003o0000o`0000H0 o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0 o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003o0000o`0000H0 o`0000<000000?l0003o00001@3o00000`000000oooo0?ooo`0:0?ooo`003`3oool00`000000oooo 0?ooo`1B0?ooo`X000001`3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`00 00<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003o0000o`0000H0o`00 00<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`00 00<000000?ooo`3oool02P3oool000l0oooo00<000000?ooo`3oool0F`3oool3000000D0o`0000<0 00000?l0003o00001P3o00000`000000o`000?l000050?l000030000003o0000o`0000H0o`0000<0 00000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<0 00000?l0003o00001P3o00000`000000o`000?l000050?l000030000003oool0oooo00X0oooo0008 0?ooo`8000001@3oool00`000000oooo0?ooo`1K0?ooo`030000003o0000o`00008000000`3o0000 0`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o0000 0`000000o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o0000 0`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool0 00L0oooo00@000000?ooo`3oool00000103oool00`000000oooo0?ooo`1K0?ooo`030000003o0000 o`000080o`0000@000000?l0003o00000000203o00000`000000o`000?l000050?l000030000003o 0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o 0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003o ool0oooo00X0oooo00090?ooo`030000003oool0oooo00<0oooo0P00001L0?ooo`030000003o0000 o`0000<0o`000`0000080?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o0000 0`000000o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o0000 0`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool0 00X0oooo00<000000?ooo`3oool00P3oool00`000000oooo0?ooo`1K0?ooo`T00000203o00000`00 0000o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`00 0000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`00 0000o`000?l000050?l000030000003oool0oooo00X0oooo00070?ooo`040000003oool0oooo0000 00@0oooo00<000000?ooo`3oool0H`3oool2000000L0o`0000<000000?l0003o00001@3o00000`00 0000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`00 0000o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`00 0000oooo0?ooo`0:0?ooo`00203oool2000000D0oooo00<000000?ooo`3oool0H`3oool010000000 o`000000000000050?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`00 0000o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`00 0000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool000l0 oooo00<000000?ooo`3oool0H`3oool01@000000o`000?l0003o0000000000@0o`0000<000000?l0 003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0 003o00001P3o00000`000000o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0 003o00001@3o00000`000000oooo0?ooo`0:0?ooo`003`3oool00`000000oooo0?ooo`1S0?ooo`03 0000003o0000o`000080o`000P0000020?l000030000003o0000o`0000D0o`0000<000000?l0003o 00001P3o00000`000000o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o 00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?ooo`3o ool02P3oool000l0oooo00<000000?ooo`3oool0H`3oool00`000000o`000?l000040?l000030000 003o0000000000L0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000 003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000 003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool000l0oooo00<000000?ooo`3oool0H`3o ool:000000L0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003o 0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o 0000o`0000D0o`0000<000000?ooo`3oool02P3oool000l0oooo00<000000?ooo`3oool0K03oool3 000000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003o0000 o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000 o`0000D0o`0000<000000?ooo`3oool02P3oool000l0oooo00<000000?ooo`3oool0K03oool01000 0000o`000?l0000000040?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o0000 0`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o0000 0`000000o`000?l000050?l000030000003oool0oooo00X0oooo000?0?ooo`030000003oool0oooo 06`0oooo00D000000?l0003o0000o`00000000030?l000030000003o0000o`0000H0o`0000<00000 0?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<00000 0?l0003o00001P3o00000`000000o`000?l000050?l000030000003oool0oooo00X0oooo000?0?oo o`030000003oool0oooo06`0oooo00<000000?l0003o00000P3o000010000000o`000?l000000008 0?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l00005 0?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000oooo0?ooo`0: 0?ooo`003`3oool00`000000oooo0?ooo`1/0?ooo`030000003o0000o`0000<0o`000`0000080?l0 00030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l0 00030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000oooo0?ooo`0:0?oo o`002@3oool3000000<0oooo00<000000?ooo`3oool0K03oool9000000P0o`0000<000000?l0003o 00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o 00001P3o00000`000000o`000?l000050?l000030000003oool0oooo00X0oooo000:0?ooo`030000 003oool0oooo0080oooo00<000000?ooo`3oool0M03oool2000000L0o`0000<000000?l0003o0000 1@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o0000 1P3o00000`000000o`000?l000050?l000030000003oool0oooo00X0oooo00070?ooo`D000000`3o ool2000007D0oooo00<000000?l0000000001P3o00000`000000o`000?l000050?l000030000003o 0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o 0000o`0000D0o`0000<000000?ooo`3oool02P3oool000L0oooo00@000000?ooo`3oool00000103o ool00`000000oooo0?ooo`1d0?ooo`030000003o0000o`0000800000103o00000`000000o`000?l0 00050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000o`000?l0 00060?l000030000003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool000P0oooo00<00000 0?ooo`000000103oool00`000000oooo0?ooo`1d0?ooo`030000003o0000o`000080o`0000D00000 0?l0003o0000o`00000000070?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o 00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?ooo`3oool02P3o ool000T0oooo0P0000040?ooo`030000003oool0oooo07@0oooo00<000000?l0003o00000`3o0000 10000000o`000?l0000000070?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o 00000`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?ooo`3oool02P3o ool000l0oooo00<000000?ooo`3oool0M03oool00`000000o`000?l000040?l000<000001`3o0000 0`000000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o0000 0`000000o`000?l000050?l000030000003oool0oooo00X0oooo000?0?ooo`030000003oool0oooo 07@0oooo2P0000070?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`00 0000o`000?l000060?l000030000003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool000l0 oooo00<000000?ooo`3oool0O@3oool2000000H0o`0000<000000?l0003o00001P3o00000`000000 o`000?l000050?l000030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000 oooo0?ooo`0:0?ooo`003`3oool00`000000oooo0?ooo`1m0?ooo`030000003o0000000000D0o`00 00<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003o0000o`0000H0o`00 00<000000?l0003o00001@3o00000`000000oooo0?ooo`0:0?ooo`003`3oool00`000000oooo0?oo o`1m0?ooo`030000003o0000o`00008000000`3o00000`000000o`000?l000060?l000030000003o 0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003o ool0oooo00X0oooo000?0?ooo`030000003oool0oooo07d0oooo00<000000?l0003o00000P3o0000 10000000o`000?l0000000080?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o 00000`000000o`000?l000050?l000030000003oool0oooo00X0oooo000?0?ooo`030000003oool0 oooo07d0oooo00<000000?l0003o00000`3o00000`000000o`00000000080?l000030000003o0000 o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003oool0 oooo00X0oooo000?0?ooo`030000003oool0oooo07d0oooo00<000000?l0003o0000103o00020000 00P0o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`00 00D0o`0000<000000?ooo`3oool02P3oool000l0oooo00<000000?ooo`3oool0O@3oool9000000P0 o`0000<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0 o`0000<000000?ooo`3oool02P3oool000P0oooo0P0000050?ooo`030000003oool0oooo08D0oooo 0P0000070?l000030000003o0000o`0000D0o`0000<000000?l0003o00001P3o00000`000000o`00 0?l000050?l000030000003oool0oooo00X0oooo00070?ooo`040000003oool0oooo000000@0oooo 00<000000?ooo`3oool0Q@3oool00`000000o`00000000060?l000030000003o0000o`0000D0o`00 00<000000?l0003o00001P3o00000`000000o`000?l000050?l000030000003oool0oooo00X0oooo 000:0?ooo`030000003oool0oooo0080oooo0P0000260?ooo`040000003o0000o`00000000D0o`00 00<000000?l0003o00001@3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`00 00<000000?ooo`3oool02P3oool000P0oooo0P0000050?ooo`030000003oool0oooo08D0oooo00D0 00000?l0003o0000o`00000000040?l000030000003o0000o`0000D0o`0000<000000?l0003o0000 1P3o00000`000000o`000?l000050?l000030000003oool0oooo00X0oooo00080?ooo`030000003o ool0oooo00@0oooo00<000000?ooo`3oool0Q@3oool00`000000o`000?l000020?l000050000003o 0000o`000?l0000000001`3o00000`000000o`000?l000060?l000030000003o0000o`0000D0o`00 00<000000?ooo`3oool02P3oool000P0oooo0`0000040?ooo`030000003oool0oooo08D0oooo00<0 00000?l0003o00000`3o000010000000o`000?l0000000070?l000030000003o0000o`0000H0o`00 00<000000?l0003o00001@3o00000`000000oooo0?ooo`0:0?ooo`003`3oool00`000000oooo0?oo o`250?ooo`030000003o0000o`0000@0o`0000<000000?l0000000001`3o00000`000000o`000?l0 00060?l000030000003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool000l0oooo00<00000 0?ooo`3oool0Q@3oool:000000L0o`0000<000000?l0003o00001P3o00000`000000o`000?l00005 0?l000030000003oool0oooo00X0oooo000?0?ooo`030000003oool0oooo08h0oooo0P0000060?l0 00030000003o0000o`0000H0o`0000<000000?l0003o00001@3o00000`000000oooo0?ooo`0:0?oo o`003`3oool00`000000oooo0?ooo`2>0?ooo`030000003o0000000000D0o`0000<000000?l0003o 00001P3o00000`000000o`000?l000050?l000030000003oool0oooo00X0oooo000?0?ooo`030000 003oool0oooo08h0oooo00@000000?l0003o00000000103o00000`000000o`000?l000060?l00003 0000003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool000l0oooo00<000000?ooo`3oool0 SP3oool01@000000o`000?l0003o0000000000<0o`0000<000000?l0003o00001P3o00000`000000 o`000?l000050?l000030000003oool0oooo00X0oooo000?0?ooo`030000003oool0oooo08h0oooo 00<000000?l0003o00000P3o000010000000o`000?l0000000080?l000030000003o0000o`0000D0 o`0000<000000?ooo`3oool02P3oool000l0oooo00<000000?ooo`3oool0SP3oool00`000000o`00 0?l000030?l000030000003o0000000000P0o`0000<000000?l0003o00001@3o00000`000000oooo 0?ooo`0:0?ooo`003`3oool00`000000oooo0?ooo`2>0?ooo`030000003o0000o`0000@0o`000P00 00080?l000030000003o0000o`0000D0o`0000<000000?ooo`3oool02P3oool000L0oooo0`000005 0?ooo`030000003oool0oooo08h0oooo2@0000080?l000030000003o0000o`0000D0o`0000<00000 0?ooo`3oool02P3oool000L0oooo00@000000?ooo`3oool00000103oool00`000000oooo0?ooo`2F 0?ooo`8000001`3o00000`000000o`000?l000050?l000030000003oool0oooo00X0oooo00070?oo o`040000003oool0oooo000000@0oooo0P00002G0?ooo`030000003o0000000000H0o`0000<00000 0?l0003o00001@3o00000`000000oooo0?ooo`0:0?ooo`001`3oool3000000D0oooo00<000000?oo o`3oool0UP3oool010000000o`000?l0000000050?l000030000003o0000o`0000D0o`0000<00000 0?ooo`3oool02P3oool000P0oooo00<000000?ooo`3oool0103oool00`000000oooo0?ooo`2F0?oo o`050000003o0000o`000?l000000000103o00000`000000o`000?l000050?l000030000003oool0 oooo00X0oooo00080?ooo`<00000103oool00`000000oooo0?ooo`2F0?ooo`030000003o0000o`00 0080o`0000D000000?l0003o0000o`00000000070?l000030000003oool0oooo00X0oooo000?0?oo o`030000003oool0oooo09H0oooo00<000000?l0003o00000`3o000010000000o`000?l000000007 0?l000030000003oool0oooo00X0oooo000?0?ooo`030000003oool0oooo09H0oooo00<000000?l0 003o0000103o0003000000L0o`0000<000000?ooo`3oool02P3oool000l0oooo00<000000?ooo`3o ool0UP3oool:000000L0o`0000<000000?ooo`3oool02P3oool000l0oooo00<000000?ooo`3oool0 W`3oool2000000H0o`0000<000000?ooo`3oool02P3oool000l0oooo00<000000?ooo`3oool0W`3o ool00`000000o`00000000050?l000030000003oool0oooo00X0oooo000?0?ooo`030000003oool0 oooo09l0oooo00@000000?l0003o00000000103o00000`000000oooo0?ooo`0:0?ooo`003`3oool0 0`000000oooo0?ooo`2O0?ooo`050000003o0000o`000?l0000000000`3o00000`000000oooo0?oo o`0:0?ooo`003`3oool00`000000oooo0?ooo`2O0?ooo`030000003o0000o`000080o`0000@00000 0?l0003o00000000303oool000l0oooo00<000000?ooo`3oool0W`3oool00`000000o`000?l00002 0?l000040000003o0000o`00000000`0oooo00090?ooo`030000003oool0oooo00<0oooo00<00000 0?ooo`3oool0W`3oool00`000000o`000?l000030?l000030000003o0000000000`0oooo00090?oo o`030000003oool0oooo00<0oooo00<000000?ooo`3oool0W`3oool00`000000o`000?l000040?l0 00800000303oool000T0oooo00<000000?ooo`3oool00`3oool200000:00oooo2@00000<0?ooo`00 2P3oool00`000000oooo0?ooo`020?ooo`030000003oool0oooo0;@0oooo00070?ooo`040000003o ool0oooo000000@0oooo00<000000?ooo`3oool0]03oool000L0oooo100000040?ooo`030000003o ool0oooo0;@0oooo00360?ooo`00\ \>"], ImageRangeCache->{{{0, 197}, {121.313, 0}} -> {0.911731, -0.745394, \ 0.00585425, 0.0663066}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RiemannRight[f, 1, 2, 20] // N\)], "Input"], Cell[BoxData[ \(2.926875`\)], "Output"] }, Open ]], Cell[BoxData[ \(Clear[n]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Sn = RiemannRight[f, 1, 2, n] // Apart\)], "Input"], Cell[BoxData[ \(11\/4 + 3\/\(4\ n\^2\) + 7\/\(2\ n\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Limit[Sn, n \[Rule] \[Infinity]]\)\(\n\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\)\)], "Input"], Cell[BoxData[ \(11\/4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Exact = \[Integral]\_1\%2 f[ x] \(\(\[DifferentialD]x\)\(\n\)\)\)], "Input"], Cell[BoxData[ \(11\/4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Exact\)\(//\)\(N\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \(2.75`\)], "Output"] }, Open ]], Cell[BoxData[{ RowBox[{" ", StyleBox[\(RiemannLeft[f_, a_, b_, n_] := \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \[Sum]\+\(i = 1\ \)\%n f[a + \((i - 1)\)*\(b - a\)\/n]*\(b - a\)\/n\), "Text"], StyleBox[ RowBox[{ StyleBox[" ", "Text"], " "}]], StyleBox[\( (*\ left\ end\ point\ Riemann\ sum\ *) \), "Text"]}], "\[IndentingNewLine]", RowBox[{" ", RowBox[{ StyleBox[\(RiemannMid[f_, a_, b_, n_]\), "Text"], StyleBox[":=", "Text"], StyleBox["\[IndentingNewLine]", "Text"], " ", StyleBox[\(\[Sum]\+\(i = 1\)\%n\(\( f[ a + \(1\/2\) \(b - a\)\/n + \((i - 1)\)*\(b - a\)\/n]\)\(*\)\(\(b - a\)\/n\)\(\ \ \ \ \)\( (*\ mid\ point\ Riemann\ sum\ *) \)\)\), "Text"]}]}]}], "Input", CellMargins->{{26, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[Integral]\_1\%5 x\ Sin[x + 1] \[DifferentialD]x\)\(//\)\(Simplify\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\( (*\ definite\ integral\ *) \)\)\)], "Input"], Cell[BoxData[ \(Cos[2] - 5\ Cos[6] - Sin[2] + Sin[6]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(F[x_] = \(\(\[Integral]x\ Sin[ x + 1] \[DifferentialD]x\)\(//\)\(Simplify\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\( (*\ indefinite\ integral\ *) \)\)\)], "Input"], Cell[BoxData[ \(\(-x\)\ Cos[1 + x] + Sin[1 + x]\)], "Output"] }, Open ]], Cell["which leads to the same by ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(F[5]\)\(-\)\(F[ 1]\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\( (*\ Fundamental\ Theorem\ of\ Calculus\ *) \)\)\)], "Input"], Cell[BoxData[ \(Cos[2] - 5\ Cos[6] - Sin[2] + Sin[6]\)], "Output"] }, Open ]], Cell["or ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(F[5] - F[1] // N\)], "Input"], Cell[BoxData[ \(\(-6.40571119482358`\)\)], "Output"] }, Open ]], Cell["To find the numerical value directly ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_1\%5 x\ Sin[x + 1] \[DifferentialD]x // N\)], "Input"], Cell[BoxData[ \(\(-6.40571119482357964`\)\)], "Output"] }, Open ]], Cell[TextData[{ "or use the numerical procedure (especially in the case when indefinite \ integral ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_a\%b\( f(x)\) \[DifferentialD]x\)]], " cannot be found)" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(FilledPlot[Sin[x] \[ExponentialE]\^\(x\^10\), {x, 0, 1}, Fills \[Rule] {GrayLevel[ .7]}];\)\(\n\)\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 0.618034 [ [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 0.0147151 0.25733 [ [.21429 .00222 -9 -9 ] [.21429 .00222 9 0 ] [.40476 .00222 -9 -9 ] [.40476 .00222 9 0 ] [.59524 .00222 -9 -9 ] [.59524 .00222 9 0 ] [.78571 .00222 -9 -9 ] [.78571 .00222 9 0 ] [.97619 .00222 -3 -9 ] [.97619 .00222 3 0 ] [.01131 .14338 -18 -4.5 ] [.01131 .14338 0 4.5 ] [.01131 .27204 -6 -4.5 ] [.01131 .27204 0 4.5 ] [.01131 .40071 -18 -4.5 ] [.01131 .40071 0 4.5 ] [.01131 .52937 -6 -4.5 ] [.01131 .52937 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .01472 m .21429 .02097 L s [(0.2)] .21429 .00222 0 1 Mshowa .40476 .01472 m .40476 .02097 L s [(0.4)] .40476 .00222 0 1 Mshowa .59524 .01472 m .59524 .02097 L s [(0.6)] .59524 .00222 0 1 Mshowa .78571 .01472 m .78571 .02097 L s [(0.8)] .78571 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(1)] .97619 .00222 0 1 Mshowa .125 Mabswid .07143 .01472 m .07143 .01847 L s .11905 .01472 m .11905 .01847 L s .16667 .01472 m .16667 .01847 L s .2619 .01472 m .2619 .01847 L s .30952 .01472 m .30952 .01847 L s .35714 .01472 m .35714 .01847 L s .45238 .01472 m .45238 .01847 L s .5 .01472 m .5 .01847 L s .54762 .01472 m .54762 .01847 L s .64286 .01472 m .64286 .01847 L s .69048 .01472 m .69048 .01847 L s .7381 .01472 m .7381 .01847 L s .83333 .01472 m .83333 .01847 L s .88095 .01472 m .88095 .01847 L s .92857 .01472 m .92857 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .02381 .14338 m .03006 .14338 L s [(0.5)] .01131 .14338 1 0 Mshowa .02381 .27204 m .03006 .27204 L s [(1)] .01131 .27204 1 0 Mshowa .02381 .40071 m .03006 .40071 L s [(1.5)] .01131 .40071 1 0 Mshowa .02381 .52937 m .03006 .52937 L s [(2)] .01131 .52937 1 0 Mshowa .125 Mabswid .02381 .04045 m .02756 .04045 L s .02381 .06618 m .02756 .06618 L s .02381 .09191 m .02756 .09191 L s .02381 .11765 m .02756 .11765 L s .02381 .16911 m .02756 .16911 L s .02381 .19485 m .02756 .19485 L s .02381 .22058 m .02756 .22058 L s .02381 .24631 m .02756 .24631 L s .02381 .29778 m .02756 .29778 L s .02381 .32351 m .02756 .32351 L s .02381 .34924 m .02756 .34924 L s .02381 .37498 m .02756 .37498 L s .02381 .42644 m .02756 .42644 L s .02381 .45218 m .02756 .45218 L s .02381 .47791 m .02756 .47791 L s .02381 .50364 m .02756 .50364 L s .02381 .55511 m .02756 .55511 L s .02381 .58084 m .02756 .58084 L s .02381 .60657 m .02756 .60657 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .01472 m .06244 .02515 L .10458 .03651 L .14415 .04714 L .18221 .05732 L .22272 .06807 L .26171 .07833 L .30316 .08912 L .34309 .09938 L .3815 .10911 L .42237 .1193 L .46172 .12896 L .49955 .1381 L .53984 .14771 L .57861 .15692 L .61984 .16685 L .65954 .17684 L .69774 .18734 L .73838 .20038 L .77751 .21628 L .79749 .22642 L .81909 .23967 L .85916 .27378 L .8791 .29782 L .90016 .33112 L .9198 .37266 L .93814 .42468 L .95804 .50267 L .97619 .60332 L s .7 g .02381 .01472 m .06244 .02515 L .10458 .03651 L .14415 .04714 L .18221 .05732 L .22272 .06807 L .26171 .07833 L .30316 .08912 L .34309 .09938 L .3815 .10911 L .42237 .1193 L .46172 .12896 L .49955 .1381 L .53984 .14771 L .57861 .15692 L .61984 .16685 L .65954 .17684 L .69774 .18734 L .73838 .20038 L .77751 .21628 L .79749 .22642 L .81909 .23967 L .85916 .27378 L .8791 .29782 L .90016 .33112 L .9198 .37266 L .93814 .42468 L .95804 .50267 L .97619 .60332 L .97619 .01472 L .02381 .01472 L F 0 g .02381 .01472 m .06244 .02515 L .10458 .03651 L .14415 .04714 L .18221 .05732 L .22272 .06807 L .26171 .07833 L .30316 .08912 L .34309 .09938 L .3815 .10911 L .42237 .1193 L .46172 .12896 L .49955 .1381 L .53984 .14771 L .57861 .15692 L .61984 .16685 L .65954 .17684 L .69774 .18734 L .73838 .20038 L .77751 .21628 L .79749 .22642 L .81909 .23967 L .85916 .27378 L .8791 .29782 L .90016 .33112 L .9198 .37266 L .93814 .42468 L .95804 .50267 L .97619 .60332 L s MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0 0 1 0.618034 MathSubStart %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 0.0147151 0.25733 [ [.21429 .00222 -9 -9 ] [.21429 .00222 9 0 ] [.40476 .00222 -9 -9 ] [.40476 .00222 9 0 ] [.59524 .00222 -9 -9 ] [.59524 .00222 9 0 ] [.78571 .00222 -9 -9 ] [.78571 .00222 9 0 ] [.97619 .00222 -3 -9 ] [.97619 .00222 3 0 ] [.01131 .14338 -18 -4.5 ] [.01131 .14338 0 4.5 ] [.01131 .27204 -6 -4.5 ] [.01131 .27204 0 4.5 ] [.01131 .40071 -18 -4.5 ] [.01131 .40071 0 4.5 ] [.01131 .52937 -6 -4.5 ] [.01131 .52937 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .01472 m .21429 .02097 L s [(0.2)] .21429 .00222 0 1 Mshowa .40476 .01472 m .40476 .02097 L s [(0.4)] .40476 .00222 0 1 Mshowa .59524 .01472 m .59524 .02097 L s [(0.6)] .59524 .00222 0 1 Mshowa .78571 .01472 m .78571 .02097 L s [(0.8)] .78571 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(1)] .97619 .00222 0 1 Mshowa .125 Mabswid .07143 .01472 m .07143 .01847 L s .11905 .01472 m .11905 .01847 L s .16667 .01472 m .16667 .01847 L s .2619 .01472 m .2619 .01847 L s .30952 .01472 m .30952 .01847 L s .35714 .01472 m .35714 .01847 L s .45238 .01472 m .45238 .01847 L s .5 .01472 m .5 .01847 L s .54762 .01472 m .54762 .01847 L s .64286 .01472 m .64286 .01847 L s .69048 .01472 m .69048 .01847 L s .7381 .01472 m .7381 .01847 L s .83333 .01472 m .83333 .01847 L s .88095 .01472 m .88095 .01847 L s .92857 .01472 m .92857 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .02381 .14338 m .03006 .14338 L s [(0.5)] .01131 .14338 1 0 Mshowa .02381 .27204 m .03006 .27204 L s [(1)] .01131 .27204 1 0 Mshowa .02381 .40071 m .03006 .40071 L s [(1.5)] .01131 .40071 1 0 Mshowa .02381 .52937 m .03006 .52937 L s [(2)] .01131 .52937 1 0 Mshowa .125 Mabswid .02381 .04045 m .02756 .04045 L s .02381 .06618 m .02756 .06618 L s .02381 .09191 m .02756 .09191 L s .02381 .11765 m .02756 .11765 L s .02381 .16911 m .02756 .16911 L s .02381 .19485 m .02756 .19485 L s .02381 .22058 m .02756 .22058 L s .02381 .24631 m .02756 .24631 L s .02381 .29778 m .02756 .29778 L s .02381 .32351 m .02756 .32351 L s .02381 .34924 m .02756 .34924 L s .02381 .37498 m .02756 .37498 L s .02381 .42644 m .02756 .42644 L s .02381 .45218 m .02756 .45218 L s .02381 .47791 m .02756 .47791 L s .02381 .50364 m .02756 .50364 L s .02381 .55511 m .02756 .55511 L s .02381 .58084 m .02756 .58084 L s .02381 .60657 m .02756 .60657 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath MathSubEnd P % End of sub-graphic % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{190, 117.375}, ImageMargins->{{43, 0}, {0, 4}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgc00<000000;>c/`2c/k<01@2c/k<00`000000/k>c0;>c/`06 0;>c/`030000002c/k<0/k>c00D0/k>c00<000000;>c/`2c/k<01@2c/k<00`000000/k>c0;>c/`05 0;>c/`030000002c/k<0/k>c00D0/k>c00<000000;>c/`2c/k<01P2c/k<00`000000/k>c0;>c/`05 0;>c/`030000002c/k<0/k>c00D0/k>c00<000000;>c/`2c/k<01@2c/k<00`000000/k>c0;>c/`05 0;>c/`030000002c/k<0/k>c00H0/k>c00<000000;>c/`2c/k<01@2c/k<00`000000/k>c0;>c/`05 0;>c/`030000002c/k<0/k>c00D0/k>c00<000000;>c/`2c/k<01@2c/k<00`000000/k>c0;>c/`06 0;>c/`030000002c/k<0/k>c00D0/k>c00<000000;>c/`2c/k<01@2c/k<00`000000oooo0?ooo`03 0?ooo`00503oool00`000000oooo0?ooo`030?ooo`<00000V`2c/k<60?ooo`00503oool00`000000 oooo0?ooo`060?ooo`@00000U`2c/k<60?ooo`00503oool2000000/0oooo0`00002D0;>c/`H0oooo 000D0?ooo`030000003oool0oooo00d0oooo1000002@0;>c/`H0oooo000D0?ooo`030000003oool0 oooo0140oooo1000002<0;>c/`H0oooo000D0?ooo`030000003oool0oooo01D0oooo1@0000270;>c /`H0oooo000D0?ooo`030000003oool0oooo01X0oooo100000230;>c/`H0oooo000D0?ooo`800000 7`3oool300000800/k>c1P3oool001@0oooo00<000000?ooo`3oool08@3oool4000007`0/k>c1P3o ool001@0oooo00<000000?ooo`3oool09@3oool3000007T0/k>c1P3oool001@0oooo00<000000?oo o`3oool0:03oool4000007D0/k>c1P3oool001@0oooo0P00000]0?ooo`<00000LP2c/k<60?ooo`00 503oool00`000000oooo0?ooo`0_0?ooo`@00000KP2c/k<60?ooo`00503oool00`000000oooo0?oo o`0c0?ooo`D00000J@2c/k<60?ooo`00503oool00`000000oooo0?ooo`0h0?ooo`D00000I02c/k<6 0?ooo`00503oool00`000000oooo0?ooo`0m0?ooo`@00000H02c/k<60?ooo`00503oool200000480 oooo0`00001M0;>c/`H0oooo000D0?ooo`030000003oool0oooo04@0oooo0`00001J0;>c/`H0oooo 00020?ooo`800000103oool2000000@0oooo0P0000040?ooo`030000003oool0oooo04L0oooo1@00 001E0;>c/`H0oooo00001@3oool000000?ooo`3oool0000000P0oooo00@000000?ooo`3oool00000 0`3oool00`000000oooo0?ooo`1<0?ooo`D00000D02c/k<60?ooo`0000D0oooo0000003oool0oooo 0000000;0?ooo`040000003oool0oooo0?ooo`800000DP3oool4000004`0/k>c1P3oool000050?oo o`000000oooo0?ooo`0000002@3oool2000000@0oooo00<000000?ooo`3oool0E@3oool5000004L0 /k>c1P3oool000050?ooo`000000oooo0?ooo`0000002@3oool00`000000oooo0?ooo`030?ooo`03 0000003oool0oooo05X0oooo1@0000120;>c/`H0oooo00020?ooo`8000002P3oool3000000<0oooo 00<000000?ooo`3oool0G`3oool4000003h0/k>c1P3oool001@0oooo00<000000?ooo`3oool0H`3o ool3000003/0/k>c1P3oool001@0oooo0P00001W0?ooo`<00000>02c/k<60?ooo`00503oool00`00 0000oooo0?ooo`1Y0?ooo`<00000=@2c/k<60?ooo`00503oool00`000000oooo0?ooo`1/0?ooo`@0 0000<@2c/k<60?ooo`00503oool00`000000oooo0?ooo`1`0?ooo`<00000;P2c/k<60?ooo`00503o ool2000007@0oooo1000000Z0;>c/`H0oooo000D0?ooo`030000003oool0oooo07L0oooo0`00000W 0;>c/`H0oooo000D0?ooo`030000003oool0oooo07X0oooo0P00000U0;>c/`H0oooo000D0?ooo`03 0000003oool0oooo07`0oooo0P00000S0;>c/`H0oooo000D0?ooo`800000O`3oool300000200/k>c 1P3oool001@0oooo00<000000?ooo`3oool0P@3oool2000001h0/k>c1P3oool001@0oooo00<00000 0?ooo`3oool0P`3oool00`000000/k>c0;>c/`0K0;>c/`H0oooo000D0?ooo`030000003oool0oooo 08@0oooo0P00000K0;>c/`H0oooo000D0?ooo`030000003oool0oooo08H0oooo00<000000;>c/`2c /k<0602c/k<60?ooo`00503oool2000008P0oooo00<000000;>c/`2c/k<05`2c/k<60?ooo`00503o ool00`000000oooo0?ooo`280?ooo`030000002c/k<0/k>c01H0/k>c1P3oool000d0oooo1@000002 0?ooo`030000003oool0oooo08T0oooo0P00000F0;>c/`H0oooo000?0?ooo`030000003oool0oooo 0080oooo00<000000?ooo`3oool0R`3oool00`000000/k>c0;>c/`0C0;>c/`H0oooo000?0?ooo`03 0000003oool0oooo0080oooo0P00002=0?ooo`030000002c/k<0/k>c0180/k>c1P3oool000l0oooo 00<000000?ooo`3oool00P3oool00`000000oooo0?ooo`2=0?ooo`030000002c/k<0/k>c0140/k>c 1P3oool000l0oooo00<000000?ooo`3oool00P3oool00`000000oooo0?ooo`2>0?ooo`030000002c /k<0/k>c0100/k>c1P3oool000h0oooo0P0000040?ooo`030000003oool0oooo08h0oooo00<00000 0;>c/`2c/k<0402c/k<60?ooo`00503oool00`000000oooo0?ooo`2?0?ooo`030000002c/k<0/k>c 00l0/k>c1P3oool001@0oooo0P00002A0?ooo`030000002c/k<0/k>c00h0/k>c1P3oool001@0oooo 00<000000?ooo`3oool0T@3oool00`000000/k>c0;>c/`0=0;>c/`H0oooo000D0?ooo`030000003o ool0oooo0980oooo00<000000;>c/`2c/k<0302c/k<60?ooo`00503oool00`000000oooo0?ooo`2B 0?ooo`030000002c/k<0/k>c00`0/k>c1P3oool001@0oooo0P00002D0?ooo`030000002c/k<0/k>c 00/0/k>c1P3oool001@0oooo00<000000?ooo`3oool0U03oool00`000000/k>c0;>c/`0:0;>c/`H0 oooo000D0?ooo`030000003oool0oooo09@0oooo00<000000;>c/`2c/k<02P2c/k<60?ooo`00503o ool00`000000oooo0?ooo`2E0?ooo`030000002c/k<0/k>c00T0/k>c1P3oool001@0oooo00<00000 0?ooo`3oool0U@3oool00`000000/k>c0;>c/`090;>c/`H0oooo000D0?ooo`800000UP3oool00`00 0000/k>c0;>c/`090;>c/`H0oooo000D0?ooo`030000003oool0oooo09H0oooo00<000000;>c/`2c /k<0202c/k<60?ooo`00503oool00`000000oooo0?ooo`2F0?ooo`030000002c/k<0/k>c00P0/k>c 1P3oool001@0oooo00<000000?ooo`3oool0U`3oool00`000000/k>c0;>c/`070;>c/`H0oooo000D 0?ooo`800000V03oool00`000000/k>c0;>c/`070;>c/`H0oooo000D0?ooo`030000003oool0oooo 09L0oooo00<000000;>c/`2c/k<01`2c/k<60?ooo`0000<0oooo0000000000000`0000020?ooo`80 0000103oool2000000@0oooo00<000000?ooo`3oool0V03oool00`000000/k>c0;>c/`060;>c/`H0 oooo00030?ooo`030000003oool0oooo00L0oooo00@000000?ooo`3oool000000`3oool00`000000 oooo0?ooo`2H0?ooo`030000002c/k<0/k>c00H0/k>c1P3oool000<0oooo00<000000?ooo`3oool0 2P3oool010000000oooo0?ooo`3oool2000009T0oooo00<000000;>c/`2c/k<01P2c/k<60?ooo`00 0`3oool00`000000oooo0?ooo`080?ooo`800000103oool00`000000oooo0?ooo`2I0?ooo`030000 002c/k<0/k>c00D0/k>c1P3oool000<0oooo00<000000?ooo`3oool0203oool00`000000oooo0?oo o`030?ooo`030000003oool0oooo09T0oooo00<000000;>c/`2c/k<01@2c/k<60?ooo`000P3oool2 000000X0oooo0`0000030?ooo`030000003oool0oooo09T0oooo00<000000;>c/`2c/k<01@2c/k<6 0?ooo`00503oool00`000000oooo0?ooo`2J0?ooo`030000002c/k<0/k>c00@0/k>c1P3oool001@0 oooo0P00002K0?ooo`030000002c/k<0/k>c00@0/k>c1P3oool001@0oooo00<000000?ooo`3oool0 VP3oool00`000000/k>c0;>c/`040;>c/`H0oooo000D0?ooo`030000003oool0oooo09/0oooo00<0 00000;>c/`2c/k<00`2c/k<60?ooo`00503oool00`000000oooo0?ooo`2K0?ooo`030000002c/k<0 /k>c00<0/k>c1P3oool001@0oooo0P00002L0?ooo`030000002c/k<0/k>c00<0/k>c1P3oool001@0 oooo00<000000?ooo`3oool0W03oool00`000000/k>c0;>c/`020;>c/`H0oooo000D0?ooo`030000 003oool0oooo09`0oooo00<000000;>c/`2c/k<00P2c/k<60?ooo`00503oool00`000000oooo0?oo o`2L0?ooo`030000002c/k<0/k>c0080/k>c1P3oool001@0oooo00<000000?ooo`3oool0W03oool0 0`000000/k>c0;>c/`020;>c/`H0oooo000D0?ooo`800000WP3oool010000000/k>c0;>c/`2c/k<6 0?ooo`00503oool00`000000oooo0?ooo`2M0?ooo`040000002c/k<0/k>c0;>c/`H0oooo000D0?oo o`030000003oool0oooo09d0oooo00@000000;>c/`2c/k<0/k>c1P3oool001@0oooo00<000000?oo o`3oool0WP3oool00`000000/k>c0;>c/`060?ooo`00503oool2000009l0oooo00<000000;>c/`2c /k<01P3oool001@0oooo00<000000?ooo`3oool0WP3oool00`000000/k>c0;>c/`060?ooo`00503o ool00`000000oooo0?ooo`2N0?ooo`030000002c/k<0/k>c00H0oooo000=0?ooo`@000000`3oool0 0`000000oooo0?ooo`2O0?ooo`030000002c/k<0oooo00D0oooo000=0?ooo`030000003oool0oooo 00@0oooo00<000000?ooo`3oool0W`3oool00`000000/k>c0?ooo`050?ooo`003P3oool00`000000 oooo0?ooo`030?ooo`800000X03oool00`000000/k>c0?ooo`050?ooo`003`3oool00`000000oooo 0?ooo`020?ooo`030000003oool0oooo09l0oooo00<000000;>c/`3oool01@3oool000d0oooo00@0 00000?ooo`3oool000000`3oool00`000000oooo0?ooo`2O0?ooo`030000002c/k<0oooo00D0oooo 000>0?ooo`800000103oool00`000000oooo0?ooo`2O0?ooo`030000002c/k<0oooo00D0oooo000D 0?ooo`800000X@3oool00`000000oooo0?ooo`040?ooo`00503oool00`000000oooo0?ooo`2P0?oo o`030000003oool0oooo00@0oooo000D0?ooo`030000003oool0oooo0:00oooo00<000000?ooo`3o ool0103oool001@0oooo00<000000?ooo`3oool0X03oool00`000000oooo0?ooo`040?ooo`00503o ool200000:40oooo00<000000?ooo`3oool0103oool001@0oooo00<000000?ooo`3oool0X03oool0 0`000000oooo0?ooo`040?ooo`00503oool00`000000oooo0?ooo`2Q0?ooo`030000003oool0oooo 00<0oooo000D0?ooo`030000003oool0oooo0:40oooo00<000000?ooo`3oool00`3oool001@0oooo 00<000000?ooo`3oool0X@3oool00`000000oooo0?ooo`030?ooo`00503oool200000:P0oooo000D 0?ooo`030000003oool0oooo0:L0oooo000D0?ooo`030000003oool0oooo0:L0oooo0000\ \>"], ImageRangeCache->{{{0, 189}, {116.375, 0}} -> {-0.00186397, -5.00002*^-06, \ 0.00531073, 0.00859294}, {{0.3125, 188.625}, {116.375, 0}} -> {-0.124753, \ -0.266212, 0.00609547, 0.0225594}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(NIntegrate[\[ExponentialE]\^\(x\^10\)\ Sin[x], {x, 0, 1}]\)], "Input"], Cell[BoxData[ \(0.556476014418848663`\)], "Output"] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Manual, WindowSize->{666, 661}, WindowMargins->{{86, Automatic}, {-228, Automatic}}, Magnification->1 ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1739, 51, 248, 7, 93, "Subtitle"], Cell[1990, 60, 174, 7, 65, "Subtitle"], Cell[2167, 69, 86, 3, 71, "Subsubtitle"] }, Open ]], Cell[CellGroupData[{ Cell[2290, 77, 30, 0, 64, "Subtitle"], Cell[2323, 79, 210, 5, 52, "Text"], Cell[CellGroupData[{ Cell[2558, 88, 72, 1, 49, "Input"], Cell[2633, 91, 63, 1, 45, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2733, 97, 59, 1, 49, "Input"], Cell[2795, 100, 267, 4, 62, "Output"] }, Open ]], Cell[3077, 107, 17, 0, 30, "Input"], Cell[3097, 109, 72, 1, 30, "Input"], Cell[3172, 112, 375, 7, 70, "Input"], Cell[CellGroupData[{ Cell[3572, 123, 130, 2, 51, "Input"], Cell[3705, 127, 27317, 1244, 123, 16117, 1101, "GraphicsData", "PostScript", \ "Graphics"] }, Open ]], Cell[31037, 1374, 258, 6, 71, "Text"], Cell[CellGroupData[{ Cell[31320, 1384, 139, 3, 51, "Input"], Cell[31462, 1389, 17647, 716, 124, 8481, 599, "GraphicsData", "PostScript", \ "Graphics"] }, Open ]], Cell[49124, 2108, 199, 4, 70, "Input"], Cell[49326, 2114, 105, 2, 42, "Input"], Cell[49434, 2118, 291, 5, 71, "Input"], Cell[CellGroupData[{ Cell[49750, 2127, 241, 4, 106, "Input"], Cell[49994, 2133, 39661, 944, 131, 7070, 537, "GraphicsData", "PostScript", \ "Graphics"] }, Open ]], Cell[CellGroupData[{ Cell[89692, 3082, 63, 1, 30, "Input"], Cell[89758, 3085, 43, 1, 29, "Output"] }, Open ]], Cell[89816, 3089, 41, 1, 30, "Input"], Cell[CellGroupData[{ Cell[89882, 3094, 71, 1, 30, "Input"], Cell[89956, 3097, 70, 1, 42, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[90063, 3103, 166, 3, 50, "Input"], Cell[90232, 3108, 39, 1, 42, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[90308, 3114, 108, 2, 63, "Input"], Cell[90419, 3118, 39, 1, 42, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[90495, 3124, 80, 1, 50, "Input"], Cell[90578, 3127, 39, 1, 29, "Output"] }, Open ]], Cell[90632, 3131, 999, 26, 134, "Input"], Cell[CellGroupData[{ Cell[91656, 3161, 237, 5, 42, "Input"], Cell[91896, 3168, 70, 1, 29, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[92003, 3174, 192, 3, 41, "Input"], Cell[92198, 3179, 65, 1, 29, "Output"] }, Open ]], Cell[92278, 3183, 43, 0, 33, "Text"], Cell[CellGroupData[{ Cell[92346, 3187, 199, 3, 30, "Input"], Cell[92548, 3192, 70, 1, 29, "Output"] }, Open ]], Cell[92633, 3196, 19, 0, 33, "Text"], Cell[CellGroupData[{ Cell[92677, 3200, 49, 1, 30, "Input"], Cell[92729, 3203, 56, 1, 29, "Output"] }, Open ]], Cell[92800, 3207, 53, 0, 33, "Text"], Cell[CellGroupData[{ Cell[92878, 3211, 87, 1, 42, "Input"], Cell[92968, 3214, 59, 1, 29, "Output"] }, Open ]], Cell[93042, 3218, 229, 6, 33, "Text"], Cell[CellGroupData[{ Cell[93296, 3228, 145, 2, 54, "Input"], Cell[93444, 3232, 16969, 643, 130, 7406, 521, "GraphicsData", "PostScript", \ "Graphics"] }, Open ]], Cell[CellGroupData[{ Cell[110450, 3880, 90, 1, 34, "Input"], Cell[110543, 3883, 55, 1, 29, "Output"] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)