(*^ ::[ 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, 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-4, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N18, O450, L-4, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N18, O450, R32768, L-4, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N18, O450, L-4, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N18, O450, B32768, L-4, 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 7 :[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 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 MultipleIntegrals 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] 9:0,0;186,2;203,0;208,2;228,0;447,1;464,0;472,1;476,0;520,-1; 3:5,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0;2,13,9,Times,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Initialization :[font = input; initialization; noKeepOnOnePage; preserveAspect] *) <1] :[font = text; inactive; preserveAspect] The integral should be set up horizontally, describing the region by the inequalities y <= x <= 2 - y 0 <= y <= 1 ;[s] 2:0,0;86,1;122,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Integrate[4 - x^2 - 2y^2, {y,0,1}, {x,y,2-y}] :[font = output; output; inactive; preserveAspect; endGroup] 5/2 ;[o] 5 - 2 :[font = text; inactive; preserveAspect] This was not required, but here is a view of the portion of the paraboloid that lies over the triangular region: :[font = input; preserveAspect; endGroup] ParametricPlot3D[{y+2t(1-y),y,4 - (y+2t(1-y))^2 - y^2}, {y,0,1}, {t,0,1},ViewPoint->{2.738, -1.695, 1.040}] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Problem 3 :[font = text; inactive; preserveAspect] The average value of a function f over a region R in the xy-plane is defined to be the integral of f over R divided by the area of R. For this problem let R be the semicircle x^2 + y^2 <= 1 above the x-axis. Find the average distance of the points in R to the origin, i.e., find the average value of the function f[x_,y_] = Sqrt[x^2 + y^2] over R. ;[s] 25:0,0;32,1;33,0;48,1;49,0;57,1;59,0;99,1;100,0;106,1;107,0;131,1;132,0;155,1;156,0;175,1;189,0;200,1;201,0;251,1;252,0;313,1;339,0;345,1;346,0;348,-1; 2:13,13,9,Times,0,12,0,0,0;12,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect] Solution :[font = text; inactive; preserveAspect] The integrals should be done in polar coordinates where the region is defined by, 0 <= r <= 1, 0 <= t <= Pi, and the distance function is just f[r Cos[t], r Sin[t]] == r. We must and another factor of r to the integrals for the area element r dr rt. ;[s] 11:0,0;82,1;93,0;95,1;107,0;143,1;170,0;201,1;202,0;242,1;249,0;251,-1; 2:6,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] intf = Integrate[ r * r, {r,0,1}, {t,0,Pi}] :[font = output; output; inactive; preserveAspect; endGroup] Pi/3 ;[o] Pi -- 3 :[font = text; inactive; preserveAspect] The area of R is just: ;[s] 3:0,0;12,1;13,0;23,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] area = Integrate[ r , {r,0,1}, {t,0,Pi}] :[font = output; output; inactive; preserveAspect; endGroup] Pi/2 ;[o] Pi -- 2 :[font = input; preserveAspect] So the average value of f is :[font = input; preserveAspect; startGroup] fbar = intf/area :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 2/3 ;[o] 2 - 3 :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Problem 4 :[font = text; inactive; preserveAspect] Find the centroid of the region in th xy-plane bounded by the parabola y = x^2 , and the lines y == 4 and x == 0. ;[s] 9:0,0;38,1;40,0;71,1;78,0;95,1;101,0;106,1;112,0;114,-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 coordinates of the centroid, {xbar, ybar}, are just the average values of x and y over the region. The parabola y == x^2 intersects the line y == 4 at the point {2,4}, so region is described vertically by the inequalities x^2 <= y <= 4 0 <= x <= 2 ;[s] 14:0,0;33,1;45,0;78,1;79,0;84,1;85,0;116,1;124,0;145,1;151,0;165,1;170,0;226,1;262,-1; 2:7,13,9,Times,0,12,0,0,0;7,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] region=Plot[{x^2,4}, {x,0,2}, AspectRatio->1] :[font = input; preserveAspect; startGroup] area = Integrate[1,{x,0,2},{y,x^2,4}] :[font = output; output; inactive; preserveAspect; endGroup] 16/3 ;[o] 16 -- 3 :[font = input; preserveAspect; startGroup] xbar = Integrate[x,{x,0,2},{y,x^2,4}]/area :[font = output; output; inactive; preserveAspect; endGroup] 3/4 ;[o] 3 - 4 :[font = input; preserveAspect; startGroup] ybar = Integrate[y,{x,0,2},{y,x^2,4}]/area :[font = output; output; inactive; preserveAspect; endGroup] 12/5 ;[o] 12 -- 5 :[font = input; preserveAspect; endGroup] Show[region,Graphics[Point[{3/4,12/5}]]] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Problem 5 :[font = text; inactive; preserveAspect] Find the mass and first moments about the coordinate axes of a thin square plate bounded by the lines x == 0, x == 1, y == 0, and y == 1/2 in the xy-plane if the density is delta[x_,y_] = 1 - x^2 - y^2. ;[s] 13:0,0;102,1;108,0;110,1;116,0;118,1;124,0;130,1;138,0;146,1;148,0;173,1;201,0;203,-1; 2:7,13,9,Times,0,12,0,0,0;6,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect] Solution :[font = text; inactive; preserveAspect] The total mass is the integral of the density function over R: ;[s] 3:0,0;60,1;61,0;63,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] delta[x_,y_] = 1 - x^2 - y^2; ;[s] 2:0,1;1,0;31,-1; 2:1,12,10,Courier,1,12,0,0,0;1,12,9,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup] M = Integrate[delta[x,y], {x,0,1}, {y,0,1/2}] :[font = output; output; inactive; preserveAspect; endGroup] 7/24 ;[o] 7 -- 24 :[font = text; inactive; preserveAspect] The moment about the y-axis is the average value of x*delta[x,y]: ;[s] 5:0,0;21,1;22,0;52,1;64,0;66,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] My = Integrate[x*delta[x,y], {x,0,1}, {y,0,1/2}]/M :[font = output; output; inactive; preserveAspect; endGroup] 5/14 ;[o] 5 -- 14 :[font = text; inactive; preserveAspect] The moment about the x-axis is the average value of y*delta[x,y]: ;[s] 5:0,0;21,1;22,0;52,1;64,0;66,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] My = Integrate[y*delta[x,y], {x,0,1}, {y,0,1/2}]/M :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 13/56 ;[o] 13 -- 56 ^*)