(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. 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[ 40676, 1386]*) (*NotebookOutlinePosition[ 41565, 1416]*) (* CellTagsIndexPosition[ 41521, 1412]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Calculus II Lab II.2: Arc Length & Surface Area\n", FontFamily->"Nimbus sans l"], StyleBox[" ", FontFamily->"Nimbus sans l", FontVariations->{"Underline"->False}], StyleBox["using Simpson's Rule etc", FontFamily->"Nimbus sans l"] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Avantgarde", FontSize->20, FontColor->RGBColor[1, 0, 1], FontVariations->{"Underline"->True}], Cell[TextData[StyleBox["The purpose of this lab is to reinforce your \ understanding of arc length and surface area and the integration involved in \ computing them. ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14]], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Swiss721bt"], Cell[TextData[{ StyleBox["Before continuing, enlarge this window to a size that is \ comfortable for you. In addition, you might wish to alter the magnification \ by using the ``format'' menu at the top, then selecting ``magnification'' and \ then the particular magnification you want. You might wish to take notes as \ you work through this file. Saving this file to your disk every few minutes \ is highly recommended. You should check with your instructor to see if any \ part of this lab is to be submitted for grading and find out the due-date.\n\n\ ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14, FontColor->RGBColor[0, 0, 1]], StyleBox["If you have forgotten Simpson's rule and other such numerical \ integration techniques, then you should first work through the ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14, FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox["appendix to this lab", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14, FontWeight->"Bold", FontColor->RGBColor[1, 0, 1], FontVariations->{"Underline"->True}], StyleBox[", which is available from the website from where you downloaded \ this lab (do not work through the exercises in the appendix unless you have \ time\nor your instructor explicitly informs you to do so).", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14, FontWeight->"Bold", FontVariations->{"Underline"->True}] }], "Text", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Swiss721bt", Background->GrayLevel[1]], Cell[TextData[{ StyleBox["Enter input cells by putting the cursor anywhere in the cell and \ press ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14], StyleBox["\[ShiftKey]\[KeyBar]\[EnterKey]", "KeyStroke", FontSize->24], StyleBox[" .", "KeyStroke"], StyleBox[" The only exception is the following cell, which contains some \ \"hidden\" programs.", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Swiss721bt", Background->GrayLevel[1]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["STOP!!", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->18, FontColor->RGBColor[1, 1, 0], Background->GrayLevel[0.500008]], StyleBox[" \n", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->18, FontColor->RGBColor[0, 1, 1]], StyleBox["Click the cursor on the arrowed bracket on the ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14, FontColor->RGBColor[0.054902, 0.423529, 0]], StyleBox["far", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14, FontWeight->"Plain", FontColor->RGBColor[0.509804, 0, 0.905882]], StyleBox[" ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14, FontWeight->"Plain", FontColor->RGBColor[0.509804, 0, 0.905882], Background->GrayLevel[1]], StyleBox["RIGHT", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14, FontColor->RGBColor[0.509804, 0, 0.905882], Background->GrayLevel[1]], StyleBox[" ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14, FontColor->RGBColor[0.0705882, 0.556863, 0], Background->GrayLevel[1]], StyleBox["and then", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14, FontColor->RGBColor[0, 0.482353, 0]], StyleBox[" ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14, FontColor->RGBColor[0.0705882, 0.556863, 0]], StyleBox["press", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14, FontColor->RGBColor[0.117647, 0.431373, 0]], StyleBox[" ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->18, FontColor->RGBColor[0.117647, 0.431373, 0]], StyleBox["\[ShiftKey]\[KeyBar]\[EnterKey]", "KeyStroke", FontSize->24, FontColor->RGBColor[0.00784314, 0.423529, 0]], StyleBox[".", "KeyStroke", FontSize->18, FontColor->RGBColor[0.00784314, 0.423529, 0]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Swiss721bt", FontSize->14, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["\<\ Lhs[f_,{x_,a_,b_,n_}]:=Module[ \t\t{delta,i}, \t\tdelta=N[(b-a)/n]; \t\tSum[f/.{x->a+i*delta},{i,0,n-1}]*delta//N] Rhs[f_,{x_,a_,b_,n_}]:=Module[ \t\t{delta,i}, \t\tdelta=N[(b-a)/n]; \t\tSum[f/.{x->a+i*delta},{i,1,n}]*delta//N] \t\t Trap[f_,{x_,a_,b_,n_}]:=Module[ \t\t{delta,i}, \t\tdelta=N[(b-a)/n]; \t\tSum[((f/.{x->a+i*delta})+(f/.{x->a+(i+1)delta}))/2,{i,0,n-1}]*delta//N] Mid[f_,{x_,a_,b_,n_}]:=Module[ \t\t{delta,i}, \t\tdelta=N[(b-a)/n]; \t Sum[f/.{x->(2a+(2i+1)*delta)/2},{i,0,n-1}]*delta//N] Simp[f_,{x_,a_,b_,n_}]:=Module[ \t\t{delta,i}, \t\tdelta=N[(b-a)/n]; Sum[((f/.{x->a+i*delta})/2+2*(f/.{x->(2a+(2i+1)*delta)/2})+(f/.{x->a+(i+1)*\ delta})/2), {i,0,n-1}]*(delta/3)//N] Area[f_,{x_,a_,b_}]:=Module[{p,q,r,r1,s}, \t\t \tp=Join[{{a,0}},Plot[f,{x,a,b},DisplayFunction->Identity][[1,1,1,1]], \t\t{{b,0}}]; \t\t \tq=Graphics[{Hue[.5,.2,1],Polygon[p]}]; \t \tr=Plot[f,{x,a,b},AxesOrigin->{0,0}, PlotRange->All, \ DisplayFunction->Identity]; \t \tr1=Plot[f,{x,a,b},AxesOrigin->{0,0}, PlotRange->All,AxesLabel->{x,y}, \ DisplayFunction->Identity]; \t \tShow[{r1,q,FullGraphics[r]},DisplayFunction->$DisplayFunction]; \t \t] Bhs[f_,{x_,a_,b_,n_}]:=Module[ \t\t{delta,i,pic,plt,plt1,r}, \t\tdelta=N[(b-a)/n]; \t\t \t\tpic1=Table[{{Hue[.3,.4,1],Rectangle[{a+i*delta,0},{a+(i+1)*delta,f/.{x->a+\ (i+1)*delta}}]}, \t Line[{{a+i*delta,0},{a+i*delta,f/.{x->a+(i+1)*delta}}, \t\t {a+(i+1)*delta,f/.{x->a+(i+1)*delta}},{a+(i+1)*delta,0}}]},{i,0,n-1}];\ \t\t \t\tpic11=Table[{RGBColor[0,0,1],Line[{{a+i*delta,f/.{x->a+(i+1)*delta}},{a+(\ i+1)*delta,f/.{x->a+(i+1)*delta}}}]}, \t\t {i,0,n-1}]; \t\t \t\tpic2=Table[{{Hue[.3,.4,1],Rectangle[{a+i*delta,0},{a+(i+1)*delta,f/.{x->a+\ i*delta}}]}, \t Line[{{a+i*delta,0},{a+i*delta,f/.{x->a+i*delta}}, \t\t {a+(i+1)*delta,f/.{x->a+i*delta}},{a+(i+1)*delta,0}}]},{i,0,n-1}]; \t\t \t\t \t\tpic22=Table[{RGBColor[1,0,0],Line[{{a+i*delta,f/.{x->a+i*delta}}, \t\t {a+(i+1)*delta,f/.{x->a+i*delta}}}]},{i,0,n-1}]; \t\t \t\tplt=Plot[f,{x,a,b},PlotRange->All,AxesOrigin->{0,0},DisplayFunction->\ Identity]; \t\t \t\tplt1=Plot[f,{x,a,b},PlotRange->All,AxesOrigin->{0,0},AxesLabel->{x,y}, \t\t DisplayFunction->Identity]; \t\t \t Show[plt1,Graphics[pic1],Graphics[pic2],Graphics[pic11], \t Graphics[pic22],FullGraphics[plt],DisplayFunction->$\ DisplayFunction]; \t\t \t\t(Sum[f/.{x->a+i*delta},{i,1,n}]-Sum[f/.{x->a+i*delta},{i,0,n-1}])*delta//\ N] \ \>", "Input", AspectRatioFixed->True] }, Closed]], Cell[TextData[StyleBox["If you restart this notebook at a later date, you \ will need to reenter the previous cell before you can continue successfully!\n\ \n", FontFamily->"Nimbus sans l", FontSize->14]], "Text", Background->GrayLevel[1]], Cell["Arc Length", "Subtitle", FontSize->18, FontColor->RGBColor[0, 0, 1], Background->GrayLevel[1]], Cell[TextData[{ StyleBox["Recall that the arc length of the graph of ", FontFamily->"Nimbus sans l", FontSize->14], StyleBox["y", FontFamily->"Urw palladio l", FontSize->14, FontSlant->"Italic"], StyleBox[" \[Equal] ", FontFamily->"Nimbus sans l", FontSize->14], StyleBox["f", FontFamily->"Nimbus sans l", FontSize->14, FontSlant->"Italic"], StyleBox["(", FontFamily->"Nimbus sans l", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[") of a function ", FontFamily->"Nimbus sans l", FontSize->14], StyleBox["f", FontFamily->"Nimbus sans l", FontSize->14, FontSlant->"Italic"], StyleBox[" (that satisfies certain continuity and differentiability \ conditions) from ", FontFamily->"Nimbus sans l", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[" \[Equal] ", FontFamily->"Nimbus sans l", FontSize->14], StyleBox["a ", FontFamily->"Nimbus sans l", FontSize->14, FontSlant->"Italic"], StyleBox[" to ", FontFamily->"Nimbus sans l", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[" \[Equal] b ", FontFamily->"Nimbus sans l", FontSize->14, FontSlant->"Italic"], StyleBox[" is given by the integral \n", FontFamily->"Nimbus sans l", FontSize->14], StyleBox[" ", FontFamily->"Nimbus sans l", FontSize->24, FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", StyleBox["a", FontFamily->"Urw palladio l"], StyleBox["b", FontFamily->"Urw palladio l"]], RowBox[{ SqrtBox[ RowBox[{"1", "+", RowBox[{ RowBox[{ StyleBox["f", FontFamily->"Utopia"], "'"}], SuperscriptBox[ RowBox[{"(", StyleBox["x", FontFamily->"Urw palladio l"], ")"}], "2"]}]}]], RowBox[{"\[DifferentialD]", StyleBox["x", FontFamily->"Urw palladio l"]}]}]}], TraditionalForm]], FontFamily->"Nimbus sans l", FontSize->18, FontSlant->"Italic"], ".", StyleBox["\n", FontFamily->"Nimbus sans l", FontSize->14] }], "Text"], Cell[TextData[{ StyleBox["EXAMPLE 1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->18, FontColor->RGBColor[0.866667, 0.411765, 0]], StyleBox["\n", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14], StyleBox["Find the length of the curve defined by ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox["y", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[" ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14, FontSlant->"Italic"], StyleBox["\[Equal] sin ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox["x", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[" on [0, 2\[Pi]] by using right-hand endpoints to compute \ the integral.\n", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox["Solution", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14, FontVariations->{"Underline"->True}], StyleBox["\n We need to compute ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox[" ", FontFamily->"Nimbus sans l", FontSize->24, FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", StyleBox["0", FontFamily->"Urw palladio l"], RowBox[{"2", StyleBox["\[Pi]", FontFamily->"Urw palladio l"]}]], RowBox[{ SqrtBox[ RowBox[{"1", "+", SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["cos", FontSlant->"Plain"], " ", StyleBox["x", FontFamily->"Urw palladio l"]}], StyleBox[")", FontFamily->"Urw palladio l"]}], "2"]}]], RowBox[{"\[DifferentialD]", StyleBox["x", FontFamily->"Urw palladio l"]}]}]}], TraditionalForm]], FontFamily->"Nimbus sans l", FontSize->18, FontSlant->"Italic"], ".", StyleBox["\n \[FilledSmallCircle] Using right-hand endpoints with 4 \ subintervals on the integrand, we will use the programRhs as\n follows \ (enter the following cell):", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Avantgarde"], Cell[BoxData[ \(Rhs[\@\(1 + Cos[x]\^2\), {x, 0, 2\ \ Pi, 4}]\)], "Input"], Cell[TextData[StyleBox["\[FilledSmallCircle] Alternatively, using 10 \ subintervals on the integrand, we get", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14]], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Swiss721bt"], Cell[BoxData[ \(Rhs[\@\(1 + Cos[x]\^2\), {x, 0, 2\ \ Pi, 10}]\)], "Input"], Cell[TextData[StyleBox["\[FilledSmallCircle] Whereas using 100 subintervals \ on the integrand gives", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14]], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Swiss721bt"], Cell[BoxData[ \(Rhs[\@\(1 + Cos[x]\^2\), {x, 0, 2\ \ Pi, 100}]\)], "Input"], Cell[TextData[{ StyleBox["EXAMPLE 2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->18, FontColor->RGBColor[0.866667, 0.411765, 0]], StyleBox["\n", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14], StyleBox["Find the length of the curve defined by ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox["y", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[" ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14, FontSlant->"Italic"], StyleBox["\[Equal] sin ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox["x", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[" on [0, 2\[Pi]] by using the trapezoidal rule on the \ integral.\n", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox["Solution", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14, FontVariations->{"Underline"->True}], StyleBox["\n As in Example 1, we need to compute ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox[" ", FontFamily->"Nimbus sans l", FontSize->24, FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", StyleBox["0", FontFamily->"Urw palladio l"], RowBox[{"2", StyleBox["\[Pi]", FontFamily->"Urw palladio l"]}]], RowBox[{ SqrtBox[ RowBox[{"1", "+", SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["cos", FontSlant->"Plain"], " ", StyleBox["x", FontFamily->"Urw palladio l"]}], StyleBox[")", FontFamily->"Urw palladio l"]}], "2"]}]], RowBox[{"\[DifferentialD]", StyleBox["x", FontFamily->"Urw palladio l"]}]}]}], TraditionalForm]], FontFamily->"Nimbus sans l", FontSize->18, FontSlant->"Italic"], ".", StyleBox["\n \[FilledSmallCircle] Using the trapezoidal rule with 4 \ subintervals on the integrand, we will use the\n program Trap as follows \ (enter the following cell):", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Swiss721bt"], Cell[BoxData[ \(Trap[\@\(1 + Cos[x]\^2\), {x, 0, 2\ \ Pi, 4}]\)], "Input"], Cell[TextData[StyleBox["\[FilledSmallCircle] Alternatively, using 10 \ subintervals on the integrand, we get", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14]], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Swiss721bt"], Cell[BoxData[ \(Trap[\@\(1 + Cos[x]\^2\), {x, 0, 2\ \ Pi, 10}]\)], "Input"], Cell[TextData[StyleBox["\[FilledSmallCircle] Whereas using 100 subintervals \ on the integrand gives", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14]], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Swiss721bt"], Cell[BoxData[ \(Trap[\@\(1 + Cos[x]\^2\), {x, 0, 2\ \ Pi, 100}]\)], "Input"], Cell[TextData[{ StyleBox["EXAMPLE 3", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->18, FontColor->RGBColor[0.866667, 0.411765, 0]], StyleBox["\n", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14], StyleBox["Find the length of the curve defined by ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox["y", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[" ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14, FontSlant->"Italic"], StyleBox["\[Equal] sin ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox["x", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[" on [0, 2\[Pi]] by using Simpson's rule on the integral. ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox["(If you have forgotten Simpson's rule, read the third paragraph \ at the start of this lab.)", FontSize->14], StyleBox["\n", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox["Solution", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14, FontVariations->{"Underline"->True}], StyleBox["\n As in Examples 1 & 2, we need to compute ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox[" ", FontFamily->"Nimbus sans l", FontSize->24, FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", StyleBox["0", FontFamily->"Urw palladio l"], RowBox[{"2", StyleBox["\[Pi]", FontFamily->"Urw palladio l"]}]], RowBox[{ SqrtBox[ RowBox[{"1", "+", SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["cos", FontSlant->"Plain"], " ", StyleBox["x", FontFamily->"Urw palladio l"]}], StyleBox[")", FontFamily->"Urw palladio l"]}], "2"]}]], RowBox[{"\[DifferentialD]", StyleBox["x", FontFamily->"Urw palladio l"]}]}]}], TraditionalForm]], FontFamily->"Nimbus sans l", FontSize->18, FontSlant->"Italic"], ".", StyleBox["\n \[FilledSmallCircle] Using Simpson's rule with 4 subintervals \ on the integrand, we will use the program Simp\n as follows (enter the \ following cell):", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Swiss721bt"], Cell[BoxData[ \(Simp[\@\(1 + Cos[x]\^2\), {x, 0, 2\ \ Pi, 4}]\)], "Input"], Cell[TextData[{ StyleBox[" (Recall that Simpson's rule must be used with an ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox["even", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14, FontColor->RGBColor[1, 0, 1]], StyleBox[" number of subintervals.)", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Avantgarde"], Cell[TextData[StyleBox["\[FilledSmallCircle] Alternatively, using 10 \ subintervals on the integrand, we get", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14]], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Swiss721bt"], Cell[BoxData[ \(Simp[\@\(1 + Cos[x]\^2\), {x, 0, 2\ \ Pi, 10}]\)], "Input"], Cell[TextData[StyleBox["\[FilledSmallCircle] Whereas using 100 subintervals \ on the integrand gives", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14]], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Swiss721bt"], Cell[BoxData[ \(Simp[\@\(1 + Cos[x]\^2\), {x, 0, 2\ \ Pi, 100}]\)], "Input"], Cell["\<\ Surface Area\ \>", "Subtitle", FontSize->18, FontColor->RGBColor[0, 0, 1], Background->GrayLevel[1]], Cell[TextData[{ StyleBox["Recall that the area of the surface generated by revolving about \ the ", FontFamily->"Nimbus sans l", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox["-axis the arc of the curve \n", FontFamily->"Nimbus sans l", FontSize->14], StyleBox["y", FontFamily->"Urw palladio l", FontSize->14, FontSlant->"Italic"], StyleBox[" \[Equal] ", FontFamily->"Nimbus sans l", FontSize->14], StyleBox["f", FontFamily->"Nimbus sans l", FontSize->14, FontSlant->"Italic"], StyleBox["(", FontFamily->"Nimbus sans l", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[") of a function ", FontFamily->"Nimbus sans l", FontSize->14], StyleBox["f", FontFamily->"Nimbus sans l", FontSize->14, FontSlant->"Italic"], StyleBox[" (where f' satisfies certain continuity and differentiability \ conditions) on [a, b]\nis given by the integral \n", FontFamily->"Nimbus sans l", FontSize->14], StyleBox[" ", FontFamily->"Nimbus sans l", FontSize->24, FontSlant->"Italic"], StyleBox["2", FontFamily->"Nimbus sans l", FontSize->16, FontSlant->"Italic"], StyleBox["\[Pi]", FontFamily->"Nimbus sans l", FontSize->24, FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", StyleBox["a", FontFamily->"Urw palladio l"], StyleBox["b", FontFamily->"Urw palladio l"]], RowBox[{ StyleBox[\(f(x)\), FontFamily->"Utopia"], SqrtBox[ RowBox[{"1", "+", RowBox[{ RowBox[{ StyleBox["f", FontFamily->"Utopia"], "'"}], SuperscriptBox[ RowBox[{"(", StyleBox["x", FontFamily->"Urw palladio l"], ")"}], "2"]}]}]], RowBox[{"\[DifferentialD]", StyleBox["x", FontFamily->"Urw palladio l"]}]}]}], TraditionalForm]], FontFamily->"Nimbus sans l", FontSize->18, FontSlant->"Italic"], ".", StyleBox["\n", FontFamily->"Nimbus sans l", FontSize->14] }], "Text"], Cell[TextData[{ StyleBox["EXAMPLE 4", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->18, FontColor->RGBColor[0.866667, 0.411765, 0]], StyleBox["\n", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontSize->14], StyleBox["Find the surface area generated when the graph of ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox["y", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[" ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14, FontSlant->"Italic"], StyleBox["\[Equal] sin ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox["x", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[" on [0, \[Pi]] is revolved about the ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox["-axis", FontFamily->"Nimbus sans l", FontSize->14], StyleBox[".\n", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox["Solution", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14, FontVariations->{"Underline"->True}], StyleBox["\n We need to compute ", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14], StyleBox[" ", FontFamily->"Nimbus sans l", FontSize->24, FontSlant->"Italic"], StyleBox["2", FontFamily->"Nimbus sans l", FontSize->16, FontSlant->"Italic"], StyleBox["\[Pi]", FontFamily->"Nimbus sans l", FontSize->24, FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", StyleBox["0", FontFamily->"Urw palladio l"], StyleBox["\[Pi]", FontFamily->"Urw palladio l"]], RowBox[{ RowBox[{"(", RowBox[{"sin", " ", StyleBox["x", FontFamily->"Urw palladio l"]}], ")"}], SqrtBox[ RowBox[{"1", "+", SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["cos", FontSlant->"Plain"], " ", StyleBox["x", FontFamily->"Urw palladio l"]}], StyleBox[")", FontFamily->"Urw palladio l"]}], "2"]}]], RowBox[{"\[DifferentialD]", StyleBox["x", FontFamily->"Urw palladio l"]}]}]}], TraditionalForm]], FontFamily->"Nimbus sans l", FontSize->18, FontSlant->"Italic"], ".", StyleBox["\n We will use Simpson's rule on the integrand like in Example 3. \ \n \[FilledSmallCircle] Using 4 subintervals we have (enter the following \ cell):", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Swiss721bt"], Cell[BoxData[ \(2\ \[Pi]\ Simp[\((Sin[x])\) \@\(1 + Cos[x]\^2\), {x, 0, \ Pi, 4}]\)], "Input"], Cell[TextData[{ " ", StyleBox["\[FilledSmallCircle] Using 10 subintervals we have (enter the \ following cell):", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14] }], "Text"], Cell[BoxData[ \(2\ \[Pi]\ Simp[\((Sin[x])\) \@\(1 + Cos[x]\^2\), {x, 0, \ Pi, 10}]\)], "Input"], Cell[TextData[{ " ", StyleBox["\[FilledSmallCircle] Using 100 subintervals we have (enter the \ following cell):", Evaluatable->False, CellGroupingRules->"OutputGrouping", AspectRatioFixed->True, FontFamily->"Nimbus sans l", FontSize->14] }], "Text"], Cell[BoxData[ \(2\ \[Pi]\ Simp[\((Sin[x])\) \@\(1 + Cos[x]\^2\), {x, 0, \ Pi, 100}]\)], "Input"], Cell["\<\ Exercises\ \>", "Subtitle", FontSize->18, FontColor->RGBColor[0, 0, 1], Background->GrayLevel[1]], Cell[TextData[{ StyleBox["1. Find the length of the arc of the curve ", FontSize->14], StyleBox["y", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[" \[Equal] f(", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[") on the given intervals by using the above\n \ commands/rules. (If you have forgotten Simpson's rule, etc, read the third \ paragraph at the\n start of this lab.)\n (a) f(", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[") \[Equal] 3 ", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[" + 2 on [-1, 2].\n (b) f(", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[") \[Equal] 1 - 2 ", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[" on [1, 3]. \n", FontSize->14], StyleBox[" do", FontSize->15], StyleBox["(c) f(", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[") \[Equal] ", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{\(2\/3\), SuperscriptBox[ StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], \(3/2\)]}], TraditionalForm]]], StyleBox[" + 1 on [0, 4]. ", FontSize->14], StyleBox["(s \n ", FontSize->15], StyleBox["(d) f(", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[") \[Equal] ", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{\(1\/3\), SuperscriptBox[ StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], "3"]}], TraditionalForm]]], StyleBox[" + ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`1\/4\)]], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], \(-1\)], TraditionalForm]]], StyleBox[" on [1, 4]. \n (e) f(", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[") \[Equal] ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\(\(sin\)\(\ \)\)\)]], Cell[BoxData[ FormBox[ StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], TraditionalForm]]], StyleBox[" on [0, \[Pi] ].\n (f) f(", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[") \[Equal] ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\(\(tan\)\(\ \)\)\)]], Cell[BoxData[ FormBox[ StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], TraditionalForm]]], StyleBox[" on [0, 1].", FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Swiss721bt"], Cell[TextData[{ StyleBox["2. Did you ever wonder how your calculator (or computer) \ generates the digits of \[Pi] ? This \n question will help us generate the \ digits of \[Pi]. (If you have forgotten Simpson's rule, etc,\n read the \ third paragraph at the start of this lab.)\n (a) Consider a circle with \ radius \[Equal] 1. What is the length of the circumference? \n \ What is half of it?\n (b) Set up an integral that finds the arc length \ of the top half of a circle of radius 1.", FontSize->14], "\n ", StyleBox["(c) Use Simpson's rule (see Example 3) to approximate the \ integral in (b) by using \n (i) 4 subintervals, (ii) 12 \ subintervals, (iii) 20 subintervals, (iv) 100 subintervals.\n (d) What \ can you conclude about the first several digits of \[Pi] ? \n \ What can you conclude about generating more digits of \[Pi] ?\n \n\ 3. Find the surface area generated when the graph of each function on the \ prescribed interval\n is revolved about the ", FontSize->14], StyleBox["x-", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox["axis", FontFamily->"Nimbus sans l", FontSize->14, FontSlant->"Italic"], StyleBox[". (If you have forgotten Simpson's rule, etc, read the third \ paragraph\n at the start of this lab.)", FontSize->14], StyleBox["\n ", FontColor->RGBColor[1, 0, 1]], StyleBox["(a) f(", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[") \[Equal] 2 ", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[" + 1 on [0, 2].\n (b) f(", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[") \[Equal] ", FontSize->14], Cell[BoxData[ FormBox[ SqrtBox[ StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"]], TraditionalForm]]], StyleBox[" on [2, 6]. \n", FontSize->14], StyleBox[" do", FontSize->15], StyleBox["(c) f(", FontSize->14], StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], StyleBox[") \[Equal] ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\(\(tan\)\(\ \)\)\)]], Cell[BoxData[ FormBox[ StyleBox["x", FontFamily->"Utopia", FontSize->14, FontSlant->"Italic"], TraditionalForm]]], StyleBox[" on [0, 1]. ", FontSize->14], StyleBox["(s ", FontSize->15] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Swiss721bt"], Cell[TextData[{ StyleBox["End of Lab. \nIf you wish, save your work. Quit ", FontSize->14], StyleBox["Mathematica", FontSize->14, FontSlant->"Italic"], StyleBox[".", FontSize->14] }], "Text", FormatType->TextForm, FontFamily->"Swiss721bt", FontColor->RGBColor[0, 0, 1]] }, FrontEndVersion->"5.0 for X", ScreenRectangle->{{0, 1024}, {0, 768}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{576, 342}, WindowMargins->{{182, Automatic}, {Automatic, 133}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"NeXTAutomaticEncoding" ] (******************************************************************* 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[1754, 51, 461, 14, 74, "Text", Evaluatable->False], Cell[2218, 67, 357, 9, 54, "Text", Evaluatable->False], Cell[2578, 78, 1821, 44, 274, "Text", Evaluatable->False, CellGroupingRules->"OutputGrouping"], Cell[4402, 124, 651, 20, 85, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[5078, 148, 2642, 89, 95, "Text", Evaluatable->False], Cell[7723, 239, 2529, 82, 1152, "Input"] }, Closed]], Cell[10267, 324, 245, 5, 99, "Text"], Cell[10515, 331, 106, 3, 45, "Subtitle"], Cell[10624, 336, 2516, 91, 129, "Text"], Cell[13143, 429, 3322, 108, 197, "Text", Evaluatable->False], Cell[16468, 539, 77, 1, 36, "Input"], Cell[16548, 542, 330, 9, 36, "Text", Evaluatable->False], Cell[16881, 553, 78, 1, 36, "Input"], Cell[16962, 556, 322, 9, 36, "Text", Evaluatable->False], Cell[17287, 567, 79, 1, 36, "Input"], Cell[17369, 570, 3333, 108, 175, "Text", Evaluatable->False], Cell[20705, 680, 78, 1, 36, "Input"], Cell[20786, 683, 330, 9, 36, "Text", Evaluatable->False], Cell[21119, 694, 79, 1, 36, "Input"], Cell[21201, 697, 322, 9, 36, "Text", Evaluatable->False], Cell[21526, 708, 80, 1, 36, "Input"], Cell[21609, 711, 3617, 117, 195, "Text", Evaluatable->False], Cell[25229, 830, 78, 1, 36, "Input"], Cell[25310, 833, 697, 23, 36, "Text", Evaluatable->False], Cell[26010, 858, 330, 9, 36, "Text", Evaluatable->False], Cell[26343, 869, 79, 1, 36, "Input"], Cell[26425, 872, 322, 9, 36, "Text", Evaluatable->False], Cell[26750, 883, 80, 1, 36, "Input"], Cell[26833, 886, 118, 7, 91, "Subtitle"], Cell[26954, 895, 2403, 84, 151, "Text"], Cell[29360, 981, 3910, 131, 172, "Text", Evaluatable->False], Cell[33273, 1114, 109, 2, 36, "Input"], Cell[33385, 1118, 275, 9, 36, "Text"], Cell[33663, 1129, 110, 2, 36, "Input"], Cell[33776, 1133, 276, 9, 36, "Text"], Cell[34055, 1144, 111, 2, 36, "Input"], Cell[34169, 1148, 115, 7, 91, "Subtitle"], Cell[34287, 1157, 3319, 127, 205, "Text", Evaluatable->False], Cell[37609, 1286, 2762, 85, 360, "Text", Evaluatable->False], Cell[40374, 1373, 298, 11, 54, "Text"] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)