(*********************************************************************** 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). 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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[ 9393, 253]*) (*NotebookOutlinePosition[ 10341, 284]*) (* CellTagsIndexPosition[ 10297, 280]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ \(Off[General::spell]\)], "Input", Background->RGBColor[1, 1, 0]], Cell[TextData[{ StyleBox["Math 2443: Calculus & Analytic Geometry IV", FontSize->12, FontColor->GrayLevel[0]], "\n", StyleBox["Surface Area", FontSize->24], "\n", StyleBox["by TJ Murphy", FontSize->12, FontColor->GrayLevel[0]] }], "Text", CellFrame->{{0, 0}, {0, 3}}, FontSize->24, FontColor->RGBColor[1, 0, 1]], Cell[TextData[{ "Shows the parallelograms that are used to approximate the area of a \ surface; also discusses the derivation of the surface area integral from \ these parallelograms.", StyleBox[" ", FontSlant->"Plain", FontColor->GrayLevel[0]] }], "Text", CellFrame->{{0, 0}, {0, 3}}, FontFamily->"Times"], Cell[CellGroupData[{ Cell["Setting Up the Tangent Parallelograms", "Text", CellFrame->{{0, 0}, {0, 3}}, FontSize->18, Background->RGBColor[1, 0, 1]], Cell[CellGroupData[{ Cell[TextData[{ "We start with the function ", Cell[BoxData[ \(TraditionalForm\`f(x, y) = \(-x\^2\) + \(-y\^2\) + 20\)]], "." }], "Text"], Cell[BoxData[ \(\(f[x_, y_] = \(-x^2\) - y^2 + 20; \)\)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Then we graph it, naming the graph \"surface\". And then we put the \ graph on a nice set of axes.\ \>", "Text"], Cell[BoxData[ \(surface = Plot3D[f[x, y], \n\t\t{x, \(-1\), 2.5}, {y, \(-3\), 3}, \n\t\t PlotPoints -> 40, Mesh -> False, BoxRatios -> {1, 1, 1}]; \n\n range = 5; \nviewpoint = \n\t\ \ ViewPoint -> {3.098, \ 1.116, \ 0.779}; \naxes = Graphics3D[{\n\t\t\t\ Line[{{0, 0, 0}, {range, 0, 0}}], Line[{{0, 0, 0}, {0, range, 0}}], Line[{{0, 0, 0}, {0, 0, 4.4*range}}]}]; \n labels = Graphics3D[{\n\t\t\t\ Text["\", {range, 0, 0}], \n\t\t\t Text["\", {0, range, 0}], \n\t\t\t Text["\", {0, 0, 4.6*range}]}]; axes3D = Show[axes, labels, viewpoint, \n\t\tAxes -> False, Boxed -> False]; \n\n Show[surface, axes3D, BoxRatios -> {1, 1, 1}, \n\tAxes -> False, Boxed -> False, viewpoint]; \)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ To calculate the area of the surface for some restricted domain, in \ this case the rectangle [-1,3]x[-3,3], we first divide the rectangular domain \ into subrectangles.\ \>", "Text"], Cell[BoxData[ \(linex1 = Graphics3D[\n\t\t{Line[{{\(-1\), \(-3\), 0}, {\(-1\), 3, 0}}]}]; \n linex2 = Graphics3D[\n\t\t{Line[{{0, \(-3\), 0}, {0, 3, 0}}]}]; \n linex3 = Graphics3D[\n\t\t{Line[{{1, \(-3\), 0}, {1, 3, 0}}]}]; \n linex4 = Graphics3D[\n\t\t{Line[{{2, \(-3\), 0}, {2, 3, 0}}]}]; \n linex5 = Graphics3D[\n\t\t{Line[{{3, \(-3\), 0}, {3, 3, 0}}]}]; \n\n liney1 = Graphics3D[\n\t\t{Line[{{\(-1\), \(-3\), 0}, {3, \(-3\), 0}}]}]; \nliney2 = Graphics3D[\n\t\t{Line[{{\(-1\), \(-2\), 0}, {3, \(-2\), 0}}]}]; \n liney3 = Graphics3D[\n\t\t{Line[{{\(-1\), \(-1\), 0}, {3, \(-1\), 0}}]}]; \nliney4 = Graphics3D[\n\t\t{Line[{{\(-1\), 0, 0}, {3, 0, 0}}]}]; \n liney5 = Graphics3D[\n\t\t{Line[{{\(-1\), 1, 0}, {3, 1, 0}}]}]; \n liney6 = Graphics3D[\n\t\t{Line[{{\(-1\), 2, 0}, {3, 2, 0}}]}]; \n liney7 = Graphics3D[\n\t\t{Line[{{\(-1\), 3, 0}, {3, 3, 0}}]}]; \n\n Show[linex1, linex2, linex3, linex4, \n\tlinex5, liney1, liney2, liney3, liney4, \n\tliney5, liney6, liney7, axes3D, \n\tBoxRatios -> {1, 1, 1}, viewpoint, \n\tAxes -> False, Boxed -> False]; \n\n Show[surface, \n\tlinex1, linex2, linex3, linex4, \n\tlinex5, liney1, liney2, liney3, liney4, \n\tliney5, liney6, liney7, axes3D, \n\t BoxRatios -> {1, 1, 1}, viewpoint, \n\tAxes -> False, Boxed -> False]; \)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Then we select a point in each subrectangle, in this case we chose \ the corner closest to the origin. For each of these points, we find the \ tangent plane to f at that point, and then restrict the domain of the tangent \ plane to the rectangle that the point came from. Two examples of these \ restricted tangent planes are shown in the graph below.\ \>", "Text"], Cell[BoxData[ \(point1x = 1; \npoint1y = 1; \npoint2x = 0; \npoint2y = 2; \n\n rectangle1 = Plot3D[0, \n\t\t{x, point1x, point1x + 1}, \n \t\t{y, point1y, point1y + 1}]; \n rectangle2 = Plot3D[0, \n\t\t{x, point2x, point2x + 1}, \n \t\t{y, point2y, point2y + 1}]; \n\nderfx[x_, y_] = D[f[x, y], x]; \n derfy[x_, y_] = D[f[x, y], y]; \n\n tanplane1 = Plot3D[f[point1x, point1y] + derfx[point1x, point1y] \((x - point1x)\) + derfy[point1x, point1y] \((y - point1y)\), \n \t\t\ \ \ {x, point1x, point1x + 1}, \n \t\t\ \ \ {y, point1y, point1y + 1}, \n\t\ \ \ \ \t PlotRange -> {\(-1\), 20}, \n\t\ \ \ \ \tBoxed -> False, Axes -> False]; \n tanplane2 = Plot3D[f[point2x, point2y] + derfx[point2x, point2y] \((x - point2x)\) + derfy[point2x, point2y] \((y - point2y)\), \n \t\t\ \ \ {x, point2x, point2x + 1}, \n \t\t\ \ \ {y, point2y, point2y + 1}, \n\t\t\ \ \ \ PlotRange -> {\(-1\), 20}, \n\t\ \ \ \ \tBoxed -> False, Axes -> False]; \n\n invisibleSurface = Show[surface, \n\t\tAmbientLight -> GrayLevel[1]]; \n\n Show[invisibleSurface, \n\tlinex1, linex2, linex3, linex4, \n\t\ linex5, liney1, liney2, liney3, liney4, \n\t\ liney5, liney6, liney7, rectangle1, \n\t\ rectangle2, \ tanplane1, tanplane2, \n\ \taxes3D, viewpoint, Axes -> False, \n\t\ Boxed -> False]; \n\n Show[surface, \n\t\ linex1, linex2, linex3, linex4, \n\t\ linex5, liney1, liney2, liney3, liney4, \n\t\ liney5, liney6, liney7, rectangle1, \n\t\ rectangle2, \ tanplane1, tanplane2, \n\ \taxes3D, viewpoint, Axes -> False, \n\t\ Boxed -> False]; \)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell["\<\ Note that each \"restricted tangent plane\" has the shape of a \ parallelogram.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Using the Tangent Parallelograms to Calculate the Surface Area\ \>", "Text", CellFrame->{{0, 0}, {0, 3}}, FontSize->18, Background->RGBColor[1, 0, 1]], Cell[TextData[{ "We can use the areas of the parallelograms to approximate the area of the \ surface. Using more and more parallelgrams gets a better and better \ approximation, ultimately leading to a double integral. ", "Pick one of the parallelograms to concentrate on. Remember that it is \ tangent to the surface f(x,y) at a point we selected earlier. Start at this \ point and draw the two vectors that lie on the adjacent sides of the \ parallelogram. (We don't show this here because ", StyleBox["Mathematica", FontSlant->"Italic"], " does not draw nice arrow heads in 3D.) The cross product of these two \ vectors gives the area of the parallelogram. That cross product reduces \ nicely and leaves us with the iterated double integral for calculating the \ area of a surface z = f(x,y): ", Cell[BoxData[ \(TraditionalForm \`A(S) = \[Integral]\(\[Integral]\(\@\(\([f\_x]\)\^2 + \([f\_y]\)\^2 + 1 \)\) \[DifferentialD]A\)\)]], " (assuming that the partial derivatives are continuous)." }], "Text"] }, Closed]], Cell[TextData[{ "This work is part of the Multivariable Calculus with ", StyleBox["Mathematica", FontSlant->"Italic"], " Project at the University of Oklahoma " }], "Text", CellFrame->{{0, 0}, {0, 3}}, FontSize->10] }, FrontEndVersion->"4.0 for Macintosh", ScreenRectangle->{{0, 832}, {0, 604}}, WindowToolbars->"RulerBar", CellGrouping->Manual, WindowSize->{520, 509}, WindowMargins->{{13, Automatic}, {Automatic, 14}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, MacintoshSystemPageSetup->"\<\ 00l0005X0FP000003^P;H?oCokH?B`^Z;085:0?l0@00005X0]0000003^PF`003 1@00I00200000@0200000BL?004001ONe50000000000030001000000000@0?oC of`?BaMD0200000000400000000001T1\>" ] (*********************************************************************** Cached data follows. 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