(************** Content-type: application/mathematica ************** 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). 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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[ 21760, 479]*) (*NotebookOutlinePosition[ 22450, 503]*) (* CellTagsIndexPosition[ 22406, 499]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Volumes of Solids of Revolution--The Method of \ Shells", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]]], "Subtitle", TextAlignment->Center], Cell[TextData[{ "This ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook provides the code to produce animations to illustrate the steps \ involved in the shell method for finding volumes of solids of revolution. By \ making suitable modifications, it is possible to change the region to be \ revolved. This notebook generates a region and revolves it about the y-axis. \ \n\nDocumentation within the notebook illustrate what parts of the code \ should be changed so that different regions might be used. This notebook is \ an example only: it is not intended to cover every possible region you may \ wish to use. ", StyleBox["You may have to make additional modifications to suit the \ particular region with which you are working.", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[1, 0, 0]], " By studying the code and through experimentation you should be able to \ modify the code as your example requires.\n\nBecause it takes a significant \ time to render the graphics, it is NOT RECOMMENDED that you run the program \ in class in real time unless you have a fast processor. It is easier to \ generate the graphics ahead of time and then select the frames to animate. \ The graphics consume a large amount of space so the ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook is saved without the embedded graphics.\n\nNOTE: It is possible \ to modify the code so that the axis of revolution is the x-axis.\n\n", StyleBox["Required Packages:", FontColor->RGBColor[1, 0, 0]] }], "Text", FontFamily->"Arial"], Cell[BoxData[{ \(<< Graphics`Shapes`\), "\[IndentingNewLine]", \(<< Graphics`SurfaceOfRevolution`\), "\[IndentingNewLine]", \(<< Graphics`FilledPlot`\), "\[IndentingNewLine]", \(<< Graphics`Graphics3D`\), "\[IndentingNewLine]", \(<< Graphics`Colors`\)}], "Input"], Cell[TextData[StyleBox["Turn off spelling warning (merely a convenience).", FontColor->RGBColor[1, 0, 0]]], "Text", FontFamily->"Arial"], Cell[BoxData[{ \(\(Off[General::spell1];\)\), "\[IndentingNewLine]", \(\(Off[General::spell];\)\)}], "Input"], Cell[TextData[StyleBox["Revolution axis is the y-axis.", FontColor->RGBColor[1, 0, 0]]], "Text", FontFamily->"Arial"], Cell[BoxData[ \(\(revyaxis = {0, 1};\)\)], "Input"], Cell[TextData[StyleBox["Define the functions. In this case the region is \ bounded by two curves. Representation as a function of y AND as a function \ of x are required.", FontColor->RGBColor[1, 0, 0]]], "Text", FontFamily->"Arial"], Cell[BoxData[{ \(\(f[x_] = 1/2 x;\)\), "\[IndentingNewLine]", \(\(g[x_] = \((x - 1)\)^2;\)\)}], "Input"], Cell[TextData[StyleBox["x interval: This needs to be modified specifically \ for the functions in your example.", FontColor->RGBColor[1, 0, 0]]], "Text", FontFamily->"Arial"], Cell[BoxData[ \(a = .5; b = 2;\)], "Input"], Cell[TextData[StyleBox["Set the plotting range window and the viewpoint. \ These statements need to be modified to suit the functions in your example.", FontColor->RGBColor[1, 0, 0]]], "Text", FontFamily->"Arial"], Cell[BoxData[{ \(\(plotrange3dsurf = {{\(-b\), b}, {\(-b\), b}, {\(- .5\), 1.5}};\)\), "\[IndentingNewLine]", \(\(viewpoint3d = {0, \(-3\), 1};\)\)}], "Input"], Cell[TextData[StyleBox["Plot x and y axes.", FontColor->RGBColor[1, 0, 0]]], "Text", FontFamily->"Arial"], Cell[BoxData[ \(\(axes = Show[{Graphics3D[{Thickness[ .008], Line[{{\(-b\), 0, 0}, {b, 0, 0}}]}], Graphics3D[{Thickness[ .008], Line[{{0, 0, \(-1\)}, {0, 0, 1}}]}]}, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, Lighting \[Rule] True, ImageSize \[Rule] 350, PlotRange \[Rule] plotrange3dsurf, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[TextData[StyleBox["Establish the partition (7 in this example). ", FontColor->RGBColor[1, 0, 0]]], "Text", FontFamily->"Arial"], Cell[BoxData[{ \(\(n = 7;\)\), "\[IndentingNewLine]", \(\(\[CapitalDelta]x = \((b - a)\)/n;\)\), "\[IndentingNewLine]", \(partition[i_] := Show[{Graphics3D[{Thickness[ .007], Line[{{\((a + \((i - 1)\)*\[CapitalDelta]x)\), 0, g[a + \((i - 1)\)*\[CapitalDelta]x]}, {\((a + \((i - 1)\)*\[CapitalDelta]x)\), 0, f[a + \((i - 1)\)*\[CapitalDelta]x]}}]}], Graphics3D[{Thickness[ .007], Line[{{\((a + \((i)\)*\[CapitalDelta]x)\), 0, f[a + \((i)\)*\[CapitalDelta]x]}, {\((a + i*\[CapitalDelta]x)\), 0, g[a + i*\[CapitalDelta]x]}}]}]}, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, Lighting \[Rule] True, ImageSize \[Rule] 300, PlotRange \[Rule] plotrange3d, DisplayFunction \[Rule] Identity]\), "\[IndentingNewLine]", \(partitionall := Table[partition[i], {i, 1, n}]\)}], "Input"], Cell[TextData[StyleBox["Define the \"front\" edges of the rectangles that \ generate the shells. ", FontColor->RGBColor[1, 0, 0]]], "Text", FontFamily->"Arial"], Cell[BoxData[ \(shelledge[i_, factor_] := Show[Graphics3D[{EdgeForm[Thickness[ .007]], SurfaceColor[colors[\([i]\)]], Polygon[{{factor*\((a + \((i - 1)\)*\[CapitalDelta]x)\), \(-0.000000001\), g[a + \((i - 1/2)\)*\[CapitalDelta]x]}, {factor \((a + \((i - 1)\)*\[CapitalDelta]x)\), \(-0.000000001\), f[a + \((i - 1/2)\)*\[CapitalDelta]x]}, {factor \((a + i*\[CapitalDelta]x)\), \(-0.000000001\), f[a + \((i - 1/2)\)*\[CapitalDelta]x]}, {factor \((a + i*\[CapitalDelta]x)\), \(-0.000000001\), g[a + \((i - 1/2)\)*\[CapitalDelta]x]}}]}], BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, Lighting \[Rule] True, ImageSize \[Rule] 300, PlotRange \[Rule] plotrange3d, DisplayFunction \[Rule] Identity]\)], "Input"], Cell[TextData[StyleBox["Define the surfaces of revolution. ", FontColor->RGBColor[1, 0, 0]]], "Text", FontFamily->"Arial"], Cell[BoxData[ \(surfacerevf[angle_] := SurfaceOfRevolution[f[x], {x, a, b}, {theta, 0, angle}, RevolutionAxis \[Rule] {0, 1}, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, ImageSize \[Rule] 300, PlotRange \[Rule] plotrange3dsurf, PlotPoints \[Rule] {15, 15}, Ticks \[Rule] None, Boxed \[Rule] False, Axes \[Rule] None, DisplayFunction \[Rule] Identity, PlotLabel \[Rule] StyleForm["\", FontWeight \[Rule] "\", FontSize \[Rule] 14]]\)], "Input"], Cell[BoxData[ \(surfacerevg[angle_] := SurfaceOfRevolution[g[x], {x, a, b}, {theta, 0, angle}, RevolutionAxis \[Rule] {0, 1}, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, ImageSize \[Rule] 300, PlotPoints \[Rule] {15, 15}, PlotRange \[Rule] plotrange3dsurf, Ticks \[Rule] None, Boxed \[Rule] False, Axes \[Rule] None, DisplayFunction \[Rule] Identity, PlotLabel \[Rule] StyleForm["\", FontWeight \[Rule] "\", FontSize \[Rule] 14]]\)], "Input"], Cell[BoxData[ \(surfacerevedge[angle_] := ListSurfaceOfRevolution[{{b, 0, g[b]}, {b, 0, f[b]}}, {theta, 0, angle}, RevolutionAxis \[Rule] {0, 1}, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, ImageSize \[Rule] 300, PlotRange \[Rule] plotrange3dsurf, Ticks \[Rule] None, Boxed \[Rule] False, Axes \[Rule] None, DisplayFunction \[Rule] Identity, PlotLabel \[Rule] StyleForm["\", FontWeight \[Rule] "\", FontSize \[Rule] 14]]\)], "Input"], Cell[TextData[StyleBox["The region is illustrated using an inscribed \ 50-vertex polygon between the curves. The vertices are generated and stored \ as a list. The region is generated by constructing a polygon defined by the \ vertices.", FontColor->RGBColor[1, 0, 0]]], "Text", FontFamily->"Arial"], Cell[BoxData[{ \(\(numpts = 50;\)\), "\[IndentingNewLine]", \(\(dx = \((b - a)\)/numpts;\)\), "\[IndentingNewLine]", \(\(fpts = Table[{a + \((i - 1)\)*dx, 0, f[a + \((i - 1)\)*dx]}, {i, 1, numpts + 1}];\)\), "\n", \(\(gpts = Table[{a + \((i - 1)\)*dx, 0, g[a + \((i - 1)\)*dx]}, {i, numpts + 1, 1, \(-1\)}];\)\), "\n", \(\(allpts = Join[gpts, fpts];\)\), "\[IndentingNewLine]", \(\(regionplot = Graphics3D[{SurfaceColor[Magenta], Polygon[allpts]}, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, ImageSize \[Rule] 300, PlotRange \[Rule] plotrange3dsurf, DisplayFunction \[Rule] Identity];\)\)}], "Input"], Cell[TextData[StyleBox["Define the surfaces of revolution that construct the \ shells.", FontColor->RGBColor[1, 0, 0]]], "Text", FontFamily->"Arial"], Cell[BoxData[ \(shelltop[i_, angle_] := Show[SurfaceOfRevolution[ g[a + \((i - 1/2)\)*\[CapitalDelta]x], {x, a + \((i - 1)\)*\[CapitalDelta]x, a + i*\[CapitalDelta]x}, {theta, 0, angle}, RevolutionAxis \[Rule] {0, 1}, PlotPoints \[Rule] {15, 15}, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, ImageSize \[Rule] 300, Shading \[Rule] True, DisplayFunction \[Rule] Identity]]\)], "Input"], Cell[BoxData[ \(shellbottom[i_, angle_] := Show[SurfaceOfRevolution[ f[a + \((i - 1/2)\)*\[CapitalDelta]x], {x, a + \((i - 1)\)*\[CapitalDelta]x, a + i*\[CapitalDelta]x}, {theta, 0, angle}, RevolutionAxis \[Rule] {0, 1}, PlotPoints \[Rule] {15, 15}, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, ImageSize \[Rule] 300, Shading \[Rule] True, DisplayFunction \[Rule] Identity]]\)], "Input"], Cell[BoxData[ \(shells[i_, angle_] := Show[{ListSurfaceOfRevolution[{{a + \((i - 1)\)*\[CapitalDelta]x, 0, f[a + \((i - 1/2)\)*\[CapitalDelta]x]}, {a + \((i - 1)\)*\[CapitalDelta]x, 0, g[a + \((i - 1/2)\)*\[CapitalDelta]x]}}, {t, 0, angle}, RevolutionAxis \[Rule] {0, 1}, BoxRatios \[Rule] {3, 3, 3}, DisplayFunction \[Rule] Identity], ListSurfaceOfRevolution[{{a + i*\[CapitalDelta]x, 0, f[a + \((i - 1/2)\)*\[CapitalDelta]x]}, {a + i*\[CapitalDelta]x, 0, g[a + \((i - 1/2)\)*\[CapitalDelta]x]}}, {t, 0, angle}, RevolutionAxis \[Rule] {0, 1}, BoxRatios \[Rule] {3, 3, 3}, DisplayFunction \[Rule] Identity]}, BoxRatios \[Rule] {3, 3, 3}, DisplayFunction \[Rule] Identity, ViewPoint \[Rule] viewpoint3d, ImageSize \[Rule] 350, PlotRange \[Rule] plotrange3d]\)], "Input"], Cell[BoxData[ \(fullshells[k_, angle_] := Show[{shelltop[k, angle], shellbottom[k, angle], shells[k, angle]}, PlotRange \[Rule] {{a, b}, {\(-5\), 5}, {\(-5\), 5}}, ViewPoint \[Rule] {0, \(-2\), \(-2\)}, DisplayFunction \[Rule] Identity]\)], "Input"], Cell[TextData[StyleBox["The surfaces will be displayed as Graphics3D \ primitives that have specific colorings.", FontColor->RGBColor[1, 0, 0]]], "Text", FontFamily->"Arial"], Cell[BoxData[ \(\(colors = {Red, Blue, Goldenrod, Green, Cyan, Orange, Purple};\)\)], "Input"], Cell[TextData[StyleBox["The approximating shells are stored in a table.", FontColor->RGBColor[1, 0, 0]]], "Text", FontFamily->"Arial"], Cell[BoxData[ \(\(shelltable[angle_] := Table[Graphics3D[{SurfaceColor[colors[\([i]\)]], EdgeForm[], \(fullshells[i, angle]\)[\([1]\)]}], {i, 1, n}];\)\)], "Input"], Cell[TextData[StyleBox["The surfaces of revolution are generated by varying \ the value of theta in the SurfaceOfRevolution and ListSurfaceOfRevolution \ statements. ", FontColor->RGBColor[1, 0, 0]]], "Text", FontFamily->"Arial"], Cell[BoxData[ \(\(theta[k_] = \((k - 1)\)*2 Pi/12;\)\)], "Input"], Cell[TextData[StyleBox["The following code shows the steps (with pauses \ between the major steps) in the approximation process. 1) a partition is \ generated, 2) a typical approximating element is generated by revolving a \ rectangle about the y-axis, 3) washers are generated on each sub-interval \ with half-washers generated first, 4) the solid of revolution is displayed.", FontColor->RGBColor[1, 0, 0]]], "Text", FontFamily->"Arial"], Cell[BoxData[{ \(\(step = 3;\)\), "\[IndentingNewLine]", \(\(totalsteps = 5;\)\), "\[IndentingNewLine]", \(\(While[\((step = step + 1)\) < totalsteps; \[IndentingNewLine]Pause[ 2]; \[IndentingNewLine]Show[{axes, regionplot}, PlotLabel \[Rule] StyleForm["\", FontWeight \[Rule] "\", FontSize \[Rule] 14], BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, ImageSize \[Rule] 400, PlotRange \[Rule] plotrange3dsurf, DisplayFunction \[Rule] $DisplayFunction, Boxed \[Rule] False]; Pause[2]; \[IndentingNewLine]Pause[2]; \[IndentingNewLine]For\ [ i = 1, i < n + 1, \(i++\), Show[{axes, regionplot, Table[partition[m], {m, 1, i}]}, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, ImageSize \[Rule] 400, Lighting \[Rule] True, PlotRange \[Rule] plotrange3dsurf, PlotLabel \[Rule] StyleForm["\", FontSize \[Rule] 14\ , FontWeight \[Rule] "\"], ImageSize \[Rule] 400, DisplayFunction \[Rule] $DisplayFunction, Boxed \[Rule] False]]; Do[If[k \[Equal] 2, Show[{axes, Table[partition[m], {m, 1, n}], shelledge[4, 1], regionplot, shelledge[4, 1]}, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, ImageSize \[Rule] 400, PlotRange \[Rule] plotrange3dsurf, PlotLabel \[Rule] StyleForm["\", FontSize \[Rule] 14\ , FontWeight \[Rule] "\"\ \ ], DisplayFunction \[Rule] $DisplayFunction, Boxed \[Rule] False], \[IndentingNewLine]If[k \[Equal] 7, Show[{axes, Table[partition[m], {m, 1, n}], shelledge[4, 1], regionplot, shelledge[4, \(-1\)], shelledge[4, 1], \(shelltable[theta[k]]\)[\([4]\)]}, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, ImageSize \[Rule] 400, PlotRange \[Rule] plotrange3dsurf, PlotLabel \[Rule] StyleForm["\", FontSize \[Rule] 14\ , FontWeight \[Rule] "\"\ \ ], DisplayFunction \[Rule] $DisplayFunction, Boxed \[Rule] False], Show[{axes, Table[partition[m], {m, 1, n}], shelledge[4, 1], regionplot, shelledge[4, 1], \(shelltable[theta[k]]\)[\([4]\)]}, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, ImageSize \[Rule] 400, PlotRange \[Rule] plotrange3dsurf, PlotLabel \[Rule] StyleForm["\", FontSize \[Rule] 14\ , FontWeight \[Rule] "\"\ \ ], DisplayFunction \[Rule] $DisplayFunction, Boxed \[Rule] False]]], {k, 2, 13}]];\)\), "\[IndentingNewLine]", \(For[i = 1, i < n + 1, \(i++\), Show[{axes, partitionall, Table[{shelledge[m, 1], \(shelltable[Pi]\)[\([m]\)], shelledge[m, \(-1\)]}, {m, 1, i - 1}], shelledge[i, 1], regionplot}, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, ImageSize \[Rule] 400, Lighting \[Rule] True, PlotRange \[Rule] plotrange3dsurf, PlotLabel \[Rule] StyleForm[ StringJoin["\", ToString[i]], FontSize \[Rule] 14\ , FontWeight \[Rule] "\"], ImageSize \[Rule] 400, DisplayFunction \[Rule] $DisplayFunction, Boxed \[Rule] False]; Pause[2]; Show[{axes, partitionall, Table[{shelledge[m, 1], \(shelltable[Pi]\)[\([m]\)], shelledge[m, \(-1\)]}, {m, 1, i - 1}], shelledge[i, 1], \(shelltable[Pi]\)[\([i]\)], shelledge[i, \(-1\)], regionplot}, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, ImageSize \[Rule] 400, Lighting \[Rule] True, PlotRange \[Rule] plotrange3dsurf, PlotLabel \[Rule] StyleForm[StringJoin["\", ToString[i]], FontSize \[Rule] 14\ , FontWeight \[Rule] "\"\ \ ], DisplayFunction \[Rule] $DisplayFunction, Boxed \[Rule] False]; Pause[2]]; Pause[2];\), "\[IndentingNewLine]", \(\(For[i = 1, i < n + 1, \(i++\), \[IndentingNewLine]Show[{axes, partitionall, Table[{shelledge[k, 1], \(shelltable[Pi]\)[\([k]\)], shelledge[k, \(-1\)]}, {k, 1, n}], Table[{shelledge[m, 1], \(shelltable[2 Pi]\)[\([m]\)], shelledge[m, \(-1\)]}, {m, 1, i}], regionplot}, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, ImageSize \[Rule] 400, Lighting \[Rule] True, PlotRange \[Rule] plotrange3dsurf, PlotLabel \[Rule] StyleForm[StringJoin["\", ToString[i]], FontSize \[Rule] 14\ , FontWeight \[Rule] "\"\ \ ], DisplayFunction \[Rule] $DisplayFunction, Boxed \[Rule] False]; Pause[2]];\)\), "\[IndentingNewLine]", \(\(\(Do[ If[\ k < 3, Show[{axes, regionplot}, DisplayFunction \[Rule] $DisplayFunction, Boxed \[Rule] False, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, Lighting \[Rule] True, ImageSize \[Rule] 400, PlotRange \[Rule] plotrange3dsurf, PlotLabel \[Rule] StyleForm["\", FontSize \[Rule] 14\ , FontWeight \[Rule] "\"\ \ ]], Show[{axes, regionplot, surfacerevf[theta[k - 1]], surfacerevg[theta[k - 1]], surfacerevedge[theta[k - 1]]}, DisplayFunction \[Rule] $DisplayFunction, PlotLabel \[Rule] StyleForm["\", FontSize \[Rule] 14\ , FontWeight \[Rule] "\"\ \ ], Boxed \[Rule] False, BoxRatios \[Rule] {3, 3, 3}, ViewPoint \[Rule] viewpoint3d, Lighting \[Rule] True, ImageSize \[Rule] 400, PlotRange \[Rule] plotrange3dsurf]], {k, 2, 14}]\)\(;\)\(Pause[ 2]\)\(;\)\(\[IndentingNewLine]\)\)\)}], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " code produced for ", StyleBox["Demos with Positive Impact (NSF-DUE 9952306)", FontColor->RGBColor[1, 0, 0]], " by\n\nLila F. Roberts\nMathematics and Computer Science Department\n\ Georgia Southern University\nStatesboro, GA 30460\nlroberts@gasou.edu.\n\ Copyright 2002. All Rights Reserved." }], "Text", FontFamily->"Arial"] }, Open ]] }, FrontEndVersion->"4.1 for Microsoft Windows", ScreenRectangle->{{0, 991}, {0, 686}}, WindowSize->{983, 653}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, CellLabelAutoDelete->True, Magnification->1.25 ] (******************************************************************* 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. 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