(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. 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, M7, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-4, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R32768, L-4, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B32768, L-4, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; showRuler; currentKernel; ] :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] Math 225: Calculus III Assignment 8 :[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; Cclosed; 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. Be sure to type in comments explaining what you are doing. Remember to uncheck Show In/Out Names in the File menu and close this group before printing. ;[s] 11:0,0;188,2;205,0;207,2;222,0;227,2;247,0;466,1;483,0;491,1;495,0;539,-1; 3:6,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0;3,13,9,Times,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Initialization :[font = input; initialization; noKeepOnOnePage; preserveAspect] *) <{-0.969, 2.785, 1.659}] :[font = text; inactive; preserveAspect] The top surface is z == x, and the bottom surface is z == 0. These intersect along the x-axis, x == 0. The other equations define the region: 0 <= y <= 1 - x^2 0 <= x <= 1 ;[s] 10:0,0;19,1;25,0;53,1;59,0;87,1;88,0;95,1;101,0;142,1;180,-1; 2:5,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] volume = Integrate[1, {x,0,1}, {y,0,1-x^2}, {z,0,x}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 1/4 ;[o] 1 - 4 :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Problem 4 :[font = text; inactive; preserveAspect] Find the volume of the solid bounded below by the paraboliod z == x^2 + y^2, above by the hyperboloid z == x^2 - y^2 + 8 and on the sides by x == -2, and x == 2. ;[s] 9:0,0;61,1;75,0;103,1;121,0;142,1;149,0;155,1;161,0;163,-1; 2:5,13,9,Times,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect] Solution :[font = text; inactive; preserveAspect] The top surface is z == x^2 - y^2 + 8, and the bottom surface is z == x^2 + y^2. These intersect at: z == x^2 + y^2 == x^2 - y^2 + 8 or y^2 == 4 which represents two lines y == -2, y == 2. The boundary of the region R is defined by the other equationsx == -2, and x == 2 and these two lines, so R is a rectangle. ;[s] 19:0,0;19,1;37,0;65,1;79,0;101,1;137,0;140,1;148,0;176,1;197,0;224,1;225,0;259,1;266,0;272,1;278,0;303,1;304,0;321,-1; 2:10,13,9,Times,0,12,0,0,0;9,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] volume = Integrate[1,{x,-2,2}, {y,-2,2}, {z,x^2+y^2,x^2-y^2+8}] :[font = output; output; inactive; preserveAspect; endGroup] 256/3 ;[o] 256 --- 3 :[font = text; inactive; preserveAspect] Here's a picture (not required): :[font = input; preserveAspect] top=Plot3D[x^2-y^2+8, {x,-2,2}, {y,-2,2}]; bot=Plot3D[x^2+y^2, {x,-2,2}, {y,-2,2}]; :[font = input; preserveAspect; endGroup] Show[top,bot] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Problem 5 :[font = text; inactive; preserveAspect] Find the volume of the solid enclosed by the spherical-coordinate surface rho == 2 Sin[phi]. ;[s] 3:0,0;74,1;91,0;93,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect] Solution :[font = text; inactive; preserveAspect] In Assignment 9, Problem 5, you are asked to plot this surface in spherical coordinates. For now we need only observe that the solid is described by the inequalities 0 <= r <= 2Sin[p] 0 <= p <= Pi 0 <= t <= 2Pi The volume element in spherical coordinates is r^2 Sin[p]. ;[s] 4:0,0;166,1;223,0;270,1;282,-1; 2:2,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] volume = Integrate[ r^2 Sin[p], {t,0,2Pi}, {p,0,Pi}, {r,0,2Sin[p]}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 2*Pi^2 ;[o] 2 2 Pi ^*)