(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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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FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["t, y = sin", CellMargins->{{Inherited, 98}, {0, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["3", CellMargins->{{Inherited, 98}, {0, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["t.", CellMargins->{{Inherited, 98}, {0, Inherited}}, Evaluatable->False, AspectRatioFixed->True] }], "SmallText", CellMargins->{{Inherited, 98}, {0, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Clear[x,y,j,volume]\nx := s Cos[t]^3\ny := s Sin[t]^3\n\ndomain = \ ParametricPlot3D[{x,y,0}, {s,0,1}, {t,0,2Pi},\n\tViewPoint -> {0,0,1000}, \ Axes->{True,True,False}, \n\tPlotPoints -> {8,33}, \n\tAxesLabel -> \ {FontForm[X,{\"Times-Italic\",8}], \n\t FontForm[Y,{\"Times-Italic\ \",8}],\"\"}, \n\tDisplayFunction -> Identity];\n\nsurface = \ ParametricPlot3D[{{x,y,0},{x,y,Sqrt[1-x^2-y^2]}}, \n {s,0,1}, \ {t,0,2Pi},\n PlotPoints -> {8,33}, Boxed->False, \ ViewPoint->{1,-2,1},\n\t AxesLabel -> \ {FontForm[X,{\"Times-Italic\",8}], \n\t \ FontForm[Y,{\"Times-Italic\",8}],\"\"}, \n\t\t DisplayFunction->Identity];\n\ \nShow[GraphicsArray[{domain,surface},\n\tDisplayFunction->$DisplayFunction\n\ ]];\nj = Det[Outer[D,{x,y},{s,t}]]; \nvolume = NIntegrate[Sqrt[1-x^2-y^2] \ Abs[j], {s,0,1}, {t,0,2Pi}];\nPrint[volume \"= volume\"]", CellMargins->{{Inherited, 98}, {0, Inherited}}, AspectRatioFixed->False, FontSize->10]], "Input", CellMargins->{{Inherited, 98}, {0, Inherited}}, AspectRatioFixed->False], Cell[TextData[ "For a simple variation, try the parameter range {s,1/4,1}."], "SmallText", CellMargins->{{Inherited, Inherited}, {0, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "It is possible to adapt the procedure used for rectangular coordinates to \ the case where there are both inner and outer domain boundary curves."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Examples", CellMargins->{{Inherited, 98}, {0, Inherited}}, Evaluatable->False, PageBreakAbove->True, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " Below are examples of parametrizations for a region which is barely \ radially simple, a region which is not radially simple (choose a \ convenient point inside the curve as the center of parametrization), and a \ region enclosed by a curve described by a piecewise defined function.", CellMargins->{{Inherited, 98}, {0, Inherited}}, Evaluatable->False, PageBreakAbove->True, AspectRatioFixed->True] }], "SmallText", CellMargins->{{Inherited, 98}, {0, Inherited}}, Evaluatable->False, PageBreakAbove->True, AspectRatioFixed->True], Cell[TextData[ "Clear[x,y]\nx := 2 + 3Cos[t]\ny := 3 + 5Sin[t]\nplot1 = ParametricPlot3D[{s \ x,s y,0}, {s,0,1}, {t,0,2Pi},\n\t ViewPoint -> {0,0,1000}, \n\t \ Axes -> {True,True,False}, \n\t PlotPoints -> {10,30},\n\t \ DisplayFunction -> Identity,\n\t AxesLabel-> {FontForm[X,{\"Times-Italic\ \",8}],\n\t FontForm[Y,{\"Times-Italic\",8}],\"\"}];\n\n\ Clear[x,y]\nx := 2 + (1 - Cos[t])Cos[t]\ny := 1 + (1 - Cos[t])Sin[t]\nplot2 \ = ParametricPlot3D[{2+s((1-Cos[t])Cos[t]), 1+s((1-Cos[t])Sin[t]), 0}, \n\t \ {s,0,1}, {t,0,2Pi}, \n\t ViewPoint -> {0,0,1000},\n\t \ PlotPoints -> {10,30},\n\t Axes -> {True,True,False},\n\t \ DisplayFunction -> Identity,\n\t AxesLabel -> \ {FontForm[X,{\"Times-Italic\",8}],\n\t \ FontForm[Y,{\"Times-Italic\",8}],\" \"}];\n\nClear[x,y]\nx[t_] := If[t <= 1, \ t, 2 - t];\ny[t_] := If[t < 1, Sqrt[1 - t^2], 3Abs[t - 2] - 3];\nplot3 = \ ParametricPlot3D[{s x[t],s y[t],0}, \n {s,0,1}, {t,-1,3},\n \ ViewPoint -> {0,0,1000},\n Axes -> {True,True,False},\n \ PlotPoints -> {8,33},\n DisplayFunction -> Identity,\n \ AxesLabel -> {FontForm[X,{\"Times-Italic\",8}],\n \ FontForm[Y,{\"Times-Italic\",8}],\"\"}];\n\n\ Show[GraphicsArray[{plot1,plot2,plot3}, \n\tGraphicsSpacing -> 0.25,\n\t\ DisplayFunction->$DisplayFunction]];"], "Input", CellMargins->{{Inherited, 98}, {0, Inherited}}, LineSpacing->{1, 0}, AspectRatioFixed->False, FontFamily->"Courier", FontSize->10]}, Open]], Cell[CellGroupData[{Cell[TextData["Parametric Equations for Three-Dimensional Domains"], "Section", CellMargins->{{Inherited, 98}, {Inherited, Inherited}}, Evaluatable->False, PageBreakAbove->True, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", AspectRatioFixed->True, FontFamily->"Times", FontSize->10, FontWeight->"Plain", FontSlant->"Italic"], StyleBox[ " cannot show the parametrization of a 3D domain directly. Stacking \ parametrized surfaces is a reasonable substitute.", AspectRatioFixed->True, FontFamily->"Times", FontSize->10, FontWeight->"Plain"] }], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Example", CellMargins->{{Inherited, 98}, {0, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " Find the mass of the region bounded by the surface z = 4 + x + y + \ sin(2x), the cylinder y = x(x - 4), and the planes y = 0 and z = 0 if the \ density is z + 1. Show the 3D region of integration.", CellMargins->{{Inherited, 98}, {0, Inherited}}, Evaluatable->False, AspectRatioFixed->True] }], "SmallText", CellMargins->{{Inherited, 98}, {0, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[x,y,z,r,t,s,j]\nx := r\ny := s x(x - 4)\nz := t (4 + x + y + Sin[2x])\n\ leaves = Table[ParametricPlot3D[{x, y, z}, \n {r, 0, 4.001}, \ {s, 0, 1}, \n\t DisplayFunction -> Identity,\n\t \ PlotPoints -> {15, 15},\n \t AxesLabel -> {FontForm[X, \ {\"Times-Italic\",8}], \n\t FontForm[Y, \ {\"Times-Italic\",8}],\n\t FontForm[Z, \ {\"Times-Italic\",8}]},\n \t ViewPoint->{2,-6,4}\n \t \ ],\n \t {t, 0, 1, .2}];\n\nShow[Flatten[leaves], \n \ DisplayFunction -> $DisplayFunction,\n AspectRatio -> 1/GoldenRatio];\n\n\ density := z + 1\nj = Det[Outer[D,{x,y,z},{r,s,t}]];\nmass = \ NIntegrate[density Abs[j], {r,0,4}, {s,0,1}, {t,0,1}];\nPrint[mass \"= \ mass\"]"], "Input", CellMargins->{{Inherited, 98}, {0, Inherited}}, AspectRatioFixed->False, FontFamily->"Courier", FontSize->10], Cell[TextData[{ StyleBox[ "\nA change of variable traditionally was used to simplify either the \ integrand or the region of integration or preferably both (a small class of \ problems). 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The student's job, the \ thinking part, is to verify that the parametrization describes correctly the \ region of integration. A fringe benefit is that the region of integration now \ always has constant limits and so setting up the integral is quite routine; \ the order of integration is unimportant.\n", CellMargins->{{Inherited, 98}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True] }], "SmallText", CellMargins->{{Inherited, 98}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["With ", CellMargins->{{Inherited, 98}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", CellMargins->{{Inherited, 98}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ", parametric representation has become a feasible way of describing \ surfaces and perhaps should become a part of basic calculus. 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