(*********************************************************************** 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). 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[ 9336, 354]*) (*NotebookOutlinePosition[ 10489, 390]*) (* CellTagsIndexPosition[ 10445, 386]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData["Math 225: Calculus III Assignment 7"], "Subsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData["Name:"], "Subsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData["Section:"], "Subsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "\nI affirm that the solutions presented in this assignment are entirely my \ own work."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Signature:"], "Subsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Instructions"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This assignment contains problems involving double integrals, see Chapter \ 14 in Finney & Thomas or Chapter 4 in the Lecture Notes. Before you begin, \ you may want to review the notebooks ", StyleBox["MultipleIntegrals", FontWeight->"Bold"], " and ", StyleBox["AdvancedCalculusDemo", FontWeight->"Bold"], " in the Notebook folder or the back of the Lecture Notes. There are also \ many examples of how to do these problems in the Quizzes folder.\n\nBe sure \ to type in comments explaining what you are doing.\n\nRemember to uncheck ", StyleBox["Show In/Out Names", FontFamily->"Courier", FontWeight->"Bold"], " in the ", StyleBox["Kernel", FontFamily->"Courier", FontWeight->"Bold"], " menu and close this group before printing." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Initialization"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData["<Automatic, InitializationCell->True, AspectRatioFixed->True], Cell[TextData["i = {1, 0, 0}; j = {0, 1, 0}; k = {0, 0, 1};"], "Input", PageBreakWithin->Automatic, InitializationCell->True, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 1"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Consider the double integral\n", StyleBox["Integrate[x^3 y, {y, 0, 2}, {x, 4-y^2, 4}]", FontFamily->"Courier", FontWeight->"Bold"], "\n(The order of integration in Mathematica is from right to left so that \ in this example the integral with respect to ", StyleBox["x", FontFamily->"Courier", FontWeight->"Bold"], " is done first and then the integral with repspect to ", StyleBox["y", FontFamily->"Courier", FontWeight->"Bold"], ".) Sketch the region of integration by plotting its boundary curves and \ write an equivalent integral with the order of integration reversed. Evaluate \ both integrals and check that they agree." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 2"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Find the volume of the region bounded above by the hyperbolic sheet\n", StyleBox["z == 4 - x^2 + y^2", FontFamily->"Courier", FontWeight->"Bold"], ", below by the ", StyleBox["xy", FontFamily->"Courier", FontWeight->"Bold"], "-plane, and on the sides by the vertical planes\n", StyleBox["y == 1", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["y == -1", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 3"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "The average value of a function ", StyleBox["f", FontFamily->"Courier", FontWeight->"Bold"], " over a region ", StyleBox["R", FontFamily->"Courier", FontWeight->"Bold"], " in the ", StyleBox["xy", FontFamily->"Courier", FontWeight->"Bold"], "-plane is defined to be the integral of ", StyleBox["f", FontFamily->"Courier", FontWeight->"Bold"], " over ", StyleBox["R", FontFamily->"Courier", FontWeight->"Bold"], " divided by the area of ", StyleBox["R", FontFamily->"Courier", FontWeight->"Bold"], ". For this problem let ", StyleBox["R", FontFamily->"Courier", FontWeight->"Bold"], " be the semicircle\n", StyleBox["x^2 + y^2 <= 1", FontFamily->"Courier", FontWeight->"Bold"], " above the ", StyleBox["x", FontFamily->"Courier", FontWeight->"Bold"], "-axis. Find the average distance of the points in ", StyleBox["R", FontFamily->"Courier", FontWeight->"Bold"], " to the point ", StyleBox["{0,1}", FontFamily->"Courier", FontWeight->"Bold"], ", i.e., find the average value of the function\n", StyleBox["Sqrt[x^2 + (y-1)^2]", FontFamily->"Courier", FontWeight->"Bold"], " over ", StyleBox["R", FontFamily->"Courier", FontWeight->"Bold"], ". (Use ", StyleBox["NIntegrate[]", FontFamily->"Courier", FontWeight->"Bold"], " for the integrals)." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 4"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Find the centroid of the part of the unit disk, ", StyleBox["x^2 + y^2 <= 1", FontFamily->"Courier", FontWeight->"Bold"], ", that lies in the first quadrant. " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 5"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Find the mass and first moments about the coordinate axes of a thin \ triangular plate bounded by the lines ", StyleBox["y - x == 1", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["y + x == 1", FontFamily->"Courier", FontWeight->"Bold"], ", and ", StyleBox["y == 0", FontFamily->"Courier", FontWeight->"Bold"], " in the ", StyleBox["xy", FontFamily->"Courier", FontWeight->"Bold"], "-plane if the density is ", StyleBox["delta[x_,y_] = 1 - y^2", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 832}, {0, 604}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Manual, WindowSize->{520, 509}, WindowMargins->{{116, Automatic}, {15, Automatic}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@00000000000000060801T1T00000000000 00000000000000000000000000000000\>" ] (*********************************************************************** 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[1709, 49, 147, 3, 44, "Subsection", PageBreakWithin->Automatic], Cell[1859, 54, 115, 3, 44, "Subsection", PageBreakWithin->Automatic], Cell[1977, 59, 118, 3, 44, "Subsection", PageBreakWithin->Automatic], Cell[CellGroupData[{ Cell[2120, 66, 161, 4, 46, "Text"], Cell[2284, 72, 120, 3, 44, "Subsection", PageBreakWithin->Automatic] }, Open ]], Cell[CellGroupData[{ Cell[2441, 80, 92, 2, 44, "Subsection"], Cell[2536, 84, 851, 22, 176, "Text"], Cell[CellGroupData[{ Cell[3412, 110, 127, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[3542, 115, 130, 3, 27, "Input", PageBreakWithin->Automatic, InitializationCell->True], Cell[3675, 120, 155, 3, 27, "Input", PageBreakWithin->Automatic, InitializationCell->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[3879, 129, 122, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[4004, 134, 742, 19, 112, "Text"], Cell[4749, 155, 121, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[4873, 160, 73, 2, 27, "Input", PageBreakWithin->Automatic] }, Open ]], Cell[CellGroupData[{ Cell[4983, 167, 122, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[5108, 172, 544, 20, 64, "Text"], Cell[5655, 194, 121, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[5779, 199, 73, 2, 27, "Input", PageBreakWithin->Automatic] }, Open ]], Cell[CellGroupData[{ Cell[5889, 206, 122, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[6014, 211, 1487, 60, 99, "Text"], Cell[7504, 273, 121, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[7628, 278, 73, 2, 27, "Input", PageBreakWithin->Automatic] }, Open ]], Cell[CellGroupData[{ Cell[7738, 285, 122, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[7863, 290, 250, 8, 47, "Text"], Cell[8116, 300, 121, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[8240, 305, 73, 2, 27, "Input", PageBreakWithin->Automatic] }, Open ]], Cell[CellGroupData[{ Cell[8350, 312, 122, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[8475, 317, 645, 25, 64, "Text"], Cell[9123, 344, 121, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[9247, 349, 73, 2, 27, "Input", PageBreakWithin->Automatic] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)