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The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, N18, O450, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, N18, O450, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, N18, O450, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M18, N18, O450, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M18, N18, O450, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, N18, O450, bold, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O450, 12, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O450, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M18, N18, O450, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N18, O450, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N18, O450, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N18, O450, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N18, O450, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M18, N18, O450, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O450, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M18, N18, O450, 12, "Times"; fontset = leftheader, inactive, M18, N18, O450, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M18, N18, O450, 12, "Times"; fontset = leftfooter, inactive, M18, N18, O450, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O450, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O450, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O450, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O450, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O450, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O450, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O450, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N18, O450, 12, "Times"; paletteColors = 128; showRuler; currentKernel; ] :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] Math 225: Calculus III Assignment 9 :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] Name: :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] Section: :[font = text; inactive; preserveAspect; startGroup] I affirm that the solutions presented in this assignment are entirely my own work. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; endGroup] Signature: :[font = subsection; inactive; preserveAspect; startGroup] Instructions :[font = text; inactive; preserveAspect] This assignment contains problems involving multiple integrals, see Chapter 14 in Finney & Thomas or Chapter 4 in the Lecture Notes. Before you begin, you may want to review the notebooks MultipleIntegrals, SphericalCoords and AdvancedCalculusDemo 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. In Problem 1, Jacobian determinants can be computed using the function J[]: ?J J[f,g,{u,v}] gives the Jacobian determinant of the coordinate transformation x = f(u,v), y = g(u,v). J[f,g,h,{u,v,w}] gives the Jacobian determinant of the coordinate transformation x = f(u,v,w), y = g(u,v,w), z = h(u,v,w). To create plots in cylindrical coordinates in Problem 4, make a substitution like {x,y,z} /. {x -> r Cos[theta], y -> r Sin[theta]} and plug it into ParametricPlot3D[..,{r,a,b},{t,c,d}]. In Problem 5, use a similar procedure for spherical coordinates: {x,y,z} /. {x->rho Cos[theta]Sin[phi], y->rho Sin[theta]Sin[phi], z->rho Cos[phi]} and plug it into ParametricPlot3D[..,{theta,a,b},{phi,c,d}]. 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