(*********************************************************************** 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[ 23126, 599]*) (*NotebookOutlinePosition[ 24074, 630]*) (* CellTagsIndexPosition[ 24030, 626]*) (*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["Double Integrals", 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[{ StyleBox["Shows the graphics corresponding to an exercise in Stewart's ", FontSlant->"Plain", FontColor->GrayLevel[0]], StyleBox["Calculus ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(3rd Ed., 1995, \[Section]13.1, p.837, #4); approximates the \ volume under a surface using a double Riemann sum, then calculates the volume \ exactly using an iterated double integral.", FontSlant->"Plain", FontColor->GrayLevel[0]] }], "Text", CellFrame->{{0, 0}, {0, 3}}, FontFamily->"Times", FontColor->RGBColor[1, 0, 1]], Cell[CellGroupData[{ Cell["Preparation for the Double Riemann Sum", "Text", CellFrame->{{0, 0}, {0, 3}}, FontSize->18, Background->RGBColor[1, 0, 1]], Cell[CellGroupData[{ Cell[TextData[{ "We start with the function ", Cell[BoxData[ \(TraditionalForm\`f(x, y) = x\^2 + 4 y\)]], "." }], "Text"], Cell[BoxData[ \(\(f[x_, y_] = x^2 + 4 y; \)\)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["The following commands create nice axes.", "Text"], Cell[BoxData[ \(range = 15; \n viewpoint = \n\t\ \ ViewPoint -> {2.782, \ 1.926, \ \(-0.026\)}; axes = Graphics3D[{\n\t\t\t\ Line[{{0, 0, 0}, { .15*range, 0, 0}}], Line[{{0, 0, 0}, {0, .25*range, 0}}], Line[{{0, 0, 0}, {0, 0, range}}]}]; \n labels = Graphics3D[{\n\t\t\t\ Text["\", { .2*range, 0, 0}], Text["\", {0, .3*range, 0}], Text["\", {0, 0, 1.1*range}]}]; axes3D = Show[axes, labels, viewpoint, \n\t\tAxes -> False, Boxed -> False]; \)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Then we graph f(x,y) and name the graph \"surface\".We have \ restricted the domain of this function to the rectangle [0,2]x[0,3] which we \ also show.\ \>", "Text"], Cell[BoxData[ \(\(surface = Plot3D[f[x, y], \n\t\t{x, 0, 2}, {y, 0, 3}, \n\t\t BoxRatios -> {1, 1, 1}, Mesh -> False]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(linex2 = Graphics3D[{\n\t\t\tLine[{{2, 0, 0}, {2, 3, 0}}]}]; \n liney3 = Graphics3D[{\n\t\t\tLine[{{0, 3, 0}, {2, 3, 0}}]}]; \)], "Input",\ Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(\(Show[surface, linex2, liney3, axes3D, \n\t\tviewpoint, Boxed -> False, Axes -> False]; \)\)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Now we consider the volume of the object that lies between the \ surface and the xy-plane.\ \>", "Text"], Cell[BoxData[ \(sidex0 = ParametricPlot3D[{0, v, t*f[0, v]}, \n\t\t{v, 0, 3}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \n sidex2 = ParametricPlot3D[{2, v, t*f[2, v]}, \n\t\t{v, 0, 3}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \n sidey0 = ParametricPlot3D[{u, 0, t*f[u, 0]}, \n\t\t{u, 0, 2}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \n sidey3 = ParametricPlot3D[{u, 3, t*f[u, 3]}, \n\t\t{u, 0, 2}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(\(Animate[ Show[surface, sidex0, sidex2, sidey0, sidey3, axes3D, Axes -> False, Boxed -> False, \t ViewPoint -> {\((1 - t)\) 2.782 - 3.029 t, \((1 - t)\) 1.926 + 1.441 t, \((1 - t)\) \((\(-0.026\))\) + 0.447 t}, BoxRatios -> {1, 1, 1}], {t, 0, 1, .1}]; \)\)], "Input", AnimationDisplayTime->1.37859, Background->RGBColor[1, 1, 0]] }, Open ]], Cell[BoxData[ \(\(Show[surface, sidex0, sidex2, sidey0, sidey3, axes3D, Axes -> False, Boxed -> False, ViewPoint -> {\(-3.029\), \ \(-1.441\), \ 0.447}, BoxRatios -> {1, 1, 1}]; \)\)], "Input", AnimationDisplayTime->0.371293, Background->RGBColor[1, 1, 0]], Cell[CellGroupData[{ Cell[TextData[{ "To calculate the Riemann sum we break this rectangle into smaller \ rectangles using the partition x=1, y=1, y=2. The following set of ", StyleBox["Mathematica", FontSlant->"Italic"], " commands graphs the lines that form these subrectangles." }], "Text"], Cell[BoxData[ \(linex1 = Graphics3D[{Line[{{1, 0, 0}, {1, 3, 0}}]}]; \n liney1 = Graphics3D[{Line[{{0, 1, 0}, {2, 1, 0}}]}]; \n liney2 = Graphics3D[{Line[{{0, 2, 0}, {2, 2, 0}}]}]; \)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We turn each of the subrectangles into a \"box\" by giving a \ height. The height is determined by the function value at a point within the \ subrectangle (in this case, the middle of the rectangle). The following set \ of commands graphs the points (a,b) at which we will calculate function \ values to get the height of the corresponding boxes.\ \>", "Text"], Cell[BoxData[ \(point11 = Graphics3D[{PointSize[ .015], Point[{1/2, 1/2, 0}]}]; \n point12 = Graphics3D[{PointSize[ .015], Point[{1/2, 3/2, 0}]}]; \n point13 = Graphics3D[{PointSize[ .015], Point[{1/2, 5/2, 0}]}]; \n point21 = Graphics3D[{PointSize[ .015], Point[{3/2, 1/2, 0}]}]; \n point22 = Graphics3D[{PointSize[ .015], Point[{3/2, 3/2, 0}]}]; \n point23 = Graphics3D[{PointSize[ .015], Point[{3/2, 5/2, 0}]}]; \)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["The following are the points (a,b,f(a,b)) on the surface.", "Text"], Cell[BoxData[ \(pointf11 = Graphics3D[{PointSize[ .015], Point[{1/2, 1/2, f[1/2, 1/2]}]}]; \n pointf12 = Graphics3D[{PointSize[ .015], Point[{1/2, 3/2, f[1/2, 3/2]}]}]; \n pointf13 = Graphics3D[{PointSize[ .015], Point[{1/2, 5/2, f[1/2, 5/2]}]}]; \n pointf21 = Graphics3D[{PointSize[ .015], Point[{3/2, 1/2, f[3/2, 1/2]}]}]; \n pointf22 = Graphics3D[{PointSize[ .015], Point[{3/2, 3/2, f[3/2, 3/2]}]}]; \n pointf23 = Graphics3D[{PointSize[ .015], Point[{3/2, 5/2, f[3/2, 5/2]}]}]; \)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["Finally we display all of the graphics objects together.", "Text"], Cell[BoxData[ \(\(Show[surface, linex1, linex2, liney1, liney2, liney3, point11, point12, point13, point21, point22, point23, pointf11, pointf12, pointf13, pointf21, pointf22, pointf23, axes3D, viewpoint, Boxed -> False, Axes -> False]; \)\)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Filling the Volume with Boxes (animation)", "Text", CellFrame->{{0, 0}, {0, 3}}, FontSize->18, Background->RGBColor[1, 0, 1]], Cell["\<\ This animation shows the accumulation of six boxes used to \ approximate the volume under our surface.\ \>", "Text"], Cell[BoxData[ \(\(box11 = Show[Plot3D[f[1/2, 1/2], {x, 0, 1}, {y, 0, 1}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{0, v, w}, {v, 0, 1}, {w, 0, f[1/2, 1/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{1, v, w}, {v, 0, 1}, {w, 0, f[1/2, 1/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{u, 0, w}, {u, 0, 1}, {w, 0, f[1/2, 1/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{u, 1, w}, {u, 0, 1}, {w, 0, f[1/2, 1/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], BoxRatios -> {1, 1, 1}]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(\(box12 = Show[Plot3D[f[1/2, 3/2], {x, 0, 1}, {y, 1, 2}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{0, v, w}, {v, 1, 2}, {w, 0, f[1/2, 3/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{1, v, w}, {v, 1, 2}, {w, 0, f[1/2, 3/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{u, 1, w}, {u, 0, 1}, {w, 0, f[1/2, 3/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{u, 2, w}, {u, 0, 1}, {w, 0, f[1/2, 3/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], BoxRatios -> {1, 1, 1}]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(\(box13 = Show[Plot3D[f[1/2, 5/2], {x, 0, 1}, {y, 2, 3}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{0, v, w}, {v, 2, 3}, {w, 0, f[1/2, 5/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{1, v, w}, {v, 2, 3}, {w, 0, f[1/2, 5/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{u, 2, w}, {u, 0, 1}, {w, 0, f[1/2, 5/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{u, 3, w}, {u, 0, 1}, {w, 0, f[1/2, 5/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], BoxRatios -> {1, 1, 1}]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(\(box21 = Show[Plot3D[f[3/2, 1/2], {x, 1, 2}, {y, 0, 1}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{1, v, w}, {v, 0, 1}, {w, 0, f[3/2, 1/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{2, v, w}, {v, 0, 1}, {w, 0, f[3/2, 1/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{u, 0, w}, {u, 1, 2}, {w, 0, f[3/2, 1/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{u, 1, w}, {u, 1, 2}, {w, 0, f[3/2, 1/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], BoxRatios -> {1, 1, 1}]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(\(box22 = Show[Plot3D[f[3/2, 3/2], {x, 1, 2}, {y, 1, 2}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{1, v, w}, {v, 1, 2}, {w, 0, f[3/2, 3/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{2, v, w}, {v, 1, 2}, {w, 0, f[3/2, 3/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{u, 1, w}, {u, 1, 2}, {w, 0, f[3/2, 3/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{u, 2, w}, {u, 1, 2}, {w, 0, f[3/2, 3/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], BoxRatios -> {1, 1, 1}]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(\(box23 = Show[Plot3D[f[3/2, 5/2], {x, 1, 2}, {y, 2, 3}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{1, v, w}, {v, 2, 3}, {w, 0, f[3/2, 5/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{2, v, w}, {v, 2, 3}, {w, 0, f[3/2, 5/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{u, 2, w}, {u, 1, 2}, {w, 0, f[3/2, 5/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], \n ParametricPlot3D[{u, 3, w}, {u, 1, 2}, {w, 0, f[3/2, 5/2]}, PlotPoints -> 2, \ DisplayFunction -> Identity], BoxRatios -> {1, 1, 1}]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(Show[surface, axes3D, viewpoint, \n\tBoxed -> False, Axes -> False]; \n Show[surface, box11, axes3D, viewpoint, \n\tBoxed -> False, Axes -> False]; \n Show[surface, box11, box12, axes3D, \n\tviewpoint, Boxed -> False, \n\t Axes -> False]; \n Show[surface, box11, box12, box13, \n\taxes3D, viewpoint, Boxed -> False, \n\tAxes -> False]; \n Show[surface, box11, box12, box13, \n\tbox21, axes3D, viewpoint, \n\t Boxed -> False, Axes -> False]; \n Show[surface, box11, box12, box13, \n\tbox21, box22, axes3D, viewpoint, \n \tBoxed -> False, Axes -> False]; \n Show[surface, box11, box12, box13, \n\tbox21, box22, box23, axes3D, viewpoint, \n\tBoxed -> False, Axes -> False]; \n Animate[Show[box11, box12, box13, box21, box22, \n\tbox23, surface, axes3D, \n\t ViewPoint -> {\((1 - t)\) 2.782 - 2.515 t, \((1 - t)\) 1.926 + 2.188 t, \((1 - t)\) \((\(- .026\))\) + .580 t}, \n\tBoxed -> False, Axes -> False], {t, 0, 1, .1}]; \)], "Input", AnimationDisplayTime->748.298, Background->RGBColor[1, 1, 0]], Cell[TextData[{ "We get a \"double Riemann sum\" by adding the volume of the boxes along \ one direction (perhaps along the y-axis) first to get a sum for an entire \ \"row\" -- that's one Riemann sum. Then we add the sums for the rows together \ -- this is another Riemann sum. And we get a \"double Riemann sum\" like: ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(i = 1\)\%2\)]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(j = 1\)\%3\), StyleBox[\(\(f(a\_ij, b\_ij)\) dA\), FontSize->14]}], TraditionalForm]]], ", where dA is represents the area of the base of each box. As we take more \ and more boxes these sums are represented as integrals as we get a \"double \ integral.\"" }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Iterated Integrals (animations)", "Text", CellFrame->{{0, 0}, {0, 3}}, FontSize->18, Background->RGBColor[1, 0, 1]], Cell["\<\ Another way to fill the volume of the region is with \"slices.\" \ This animation shows the accumulation of slices parallel to the \ xz-plane.\ \>", "Text"], Cell[BoxData[ \(slicey0 = ParametricPlot3D[{u, 0, t*f[u, 0]}, \n\t\t{u, 0, 2}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \n slicey01 = ParametricPlot3D[{u, 1/2, t*f[u, 1/2]}, \n\t\t{u, 0, 2}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \n slicey1 = ParametricPlot3D[{u, 1, t*f[u, 1]}, \n\t\t{u, 0, 2}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \n slicey12 = ParametricPlot3D[{u, 3/2, t*f[u, 3/2]}, \n\t\t{u, 0, 2}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \n slicey2 = ParametricPlot3D[{u, 2, t*f[u, 2]}, \n\t\t{u, 0, 2}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \n slicey23 = ParametricPlot3D[{u, 5/2, t*f[u, 5/2]}, \n\t\t{u, 0, 2}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \n slicey3 = ParametricPlot3D[{u, 3, t*f[u, 3]}, \n\t\t{u, 0, 2}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(Show[surface, axes3D, \n\t\tviewpoint, Boxed -> False, Axes -> False]; \nShow[surface, slicey0, axes3D, \n\t\tviewpoint, Boxed -> False, Axes -> False]; \n Show[surface, slicey0, slicey01, axes3D, \n\t\tviewpoint, Boxed -> False, Axes -> False]; \n Show[surface, slicey0, slicey01, slicey1, axes3D, \n\t\tviewpoint, Boxed -> False, Axes -> False]; \n Show[surface, slicey0, slicey01, slicey1, slicey12, axes3D, \n\t\t viewpoint, Boxed -> False, Axes -> False]; \n Show[surface, slicey0, slicey01, slicey1, slicey12, slicey2, axes3D, \n \t\tviewpoint, Boxed -> False, Axes -> False]; \n Show[surface, slicey0, slicey01, slicey1, slicey12, slicey2, slicey23, axes3D, \n\t\tviewpoint, Boxed -> False, Axes -> False]; \n Show[surface, slicey0, slicey01, slicey1, slicey12, slicey2, slicey23, slicey3, axes3D, \n\t\tviewpoint, Boxed -> False, Axes -> False]; \)], "Input", AnimationDisplayTime->32.1183, Background->RGBColor[1, 1, 0]], Cell["Consider one \"slice\" in the animation:", "Text"], Cell[BoxData[ \(\(Show[surface, slicey2, axes3D, \n\t\tviewpoint, Boxed -> False, Axes -> False]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[TextData[{ "The integral ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\[Integral]\_0\%2\), ScriptLevel->0], RowBox[{ RowBox[{"(", \(x\^2 + 4 y\), StyleBox["\[ThinSpace]", ScriptLevel->0], ")"}], StyleBox[\(\[DifferentialD]x\), ScriptLevel->0]}]}], TraditionalForm]]], " calculates the area of such a slice (think about how you learned to \ calculate the area under a curve). Note that in this integral y is acting \ like a constant. Also note that if we worked out this integral, our answer \ would be in terms of y. That happens because the area of the slice depends on \ how far along the y axis we are. So if we have infinitely many of these \ slices filling up the region under the surface, we can add up their areas to \ find the volume of that region. This involves another integral: ", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%3\), RowBox[{ StyleBox[\(\[Integral]\_0\%2\), ScriptLevel->0], RowBox[{ RowBox[{"(", \(x\^2 + 4 y\), StyleBox["\[ThinSpace]", ScriptLevel->0], ")"}], StyleBox[\(\[DifferentialD]x\), ScriptLevel->0], \(\[DifferentialD]y\)}]}]}], TraditionalForm]]], ". This is called \"an iterated double integral\", in which the inner \ integral calculates the area of a slice and the outer integral adds up all \ those areas." }], "Text"], Cell["\<\ The next animation shows the accumulation of slices parallel to the \ yz-plane.\ \>", "Text"], Cell[BoxData[ \(slicex0 = ParametricPlot3D[{0, v, t*f[0, v]}, \n\t\t{v, 0, 3}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \n slicex01 = ParametricPlot3D[{1/2, v, t*f[1/2, v]}, \n\t\t{v, 0, 3}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \n slicex1 = ParametricPlot3D[{1, v, t*f[1, v]}, \n\t\t{v, 0, 3}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \n slicex12 = ParametricPlot3D[{3/2, v, t*f[3/2, v]}, \n\t\t{v, 0, 3}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \n slicex2 = ParametricPlot3D[{2, v, t*f[2, v]}, \n\t\t{v, 0, 3}, {t, 0, 1}, PlotPoints -> 10, \n\t\ \ DisplayFunction -> Identity]; \)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(Show[surface, axes3D, \n\t\tviewpoint, Boxed -> False, Axes -> False]; \nShow[surface, slicex0, axes3D, \n\t\tviewpoint, Boxed -> False, Axes -> False]; \n Show[surface, slicex0, slicex01, axes3D, \n\t\tviewpoint, Boxed -> False, Axes -> False]; \n Show[surface, slicex0, slicex01, slicex1, axes3D, \n\t\tviewpoint, Boxed -> False, Axes -> False]; \n Show[surface, slicex0, slicex01, slicex1, slicex12, axes3D, \n\t\t viewpoint, Boxed -> False, Axes -> False]; \n Show[surface, slicex0, slicex01, slicex1, slicex12, slicex2, axes3D, \n \t\tviewpoint, Boxed -> False, Axes -> False]; \)], "Input", AnimationDisplayTime->0.482681, Background->RGBColor[1, 1, 0]], Cell[TextData[{ "For this animation (with the slices going the other way), the iterated \ double integral would be: ", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%2\), RowBox[{ StyleBox[\(\[Integral]\_0\%3\), ScriptLevel->0], RowBox[{ RowBox[{"(", \(x\^2 + 4 y\), StyleBox["\[ThinSpace]", ScriptLevel->0], ")"}], StyleBox[\(\[DifferentialD]y\), ScriptLevel->0], \(\[DifferentialD]x\)}]}]}], TraditionalForm]]], ". This process of changing how we take our slices is called \"reversing \ the order of integration.\" Warning: reversing the order of integration is \ not always as straightforward as it was here!" }], "Text"], Cell[TextData[{ "To calculate this double integral using ", StyleBox["Mathematica", FontSlant->"Italic"], " you can copy and paste the expressions above, use the palettes to \ reproduce these expressions, or use the command" }], "Text"], Cell[BoxData[ \(Integrate[x^2 + 4 y, {y, 0, 3}, {x, 0, 2}]\)], "Input", Background->RGBColor[1, 1, 0]], Cell["or for the other order of integration", "Text"], Cell[BoxData[ \(Integrate[x^2 + 4 y, {x, 0, 2}, {y, 0, 3}]\)], "Input", Background->RGBColor[1, 1, 0]], Cell["\<\ to get the answer 44. Note that this example is simple enough to do \ by hand and certainly does not require the power of a computer or even a \ calculator.\ \>", "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->{{Automatic, 39}, {Automatic, 19}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, MacintoshSystemPageSetup->"\<\ 00l0005X0FP000003^P;H?oCokH?B`^Z;085:0?l0@00005X0]0000003^PF`003 1@00I00200000@0200000BL?004000]Q4n0000000000030001000000000@0?oC of`?BaMD0200000000400000000001T1\>" ] (*********************************************************************** Cached data follows. 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