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(*^ ::[paletteColors = 128; currentKernel; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = Left Header, nohscroll, cellOutline, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, L1, 12; fontset = Left Footer, cellOutline, blackBox, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; next21StandardFontEncoding; ] :[font = title; inactive; Cclosed; preserveAspect; startGroup; ] Lab 4: Solids of Revolution :[font = text; inactive; preserveAspect; ] This lab is based upon Section 8.2 in Stein. Also see pages 168-174 in the Guidebook by Crooke and Ratcliffe. :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Graphing the Surface :[font = text; inactive; Cclosed; preserveAspect; startGroup; ] First execute the following command: :[font = input; preserveAspect; ] Surface[fun_,u1:{u_,umin_,umax_}] := ParametricPlot3D[{fun Cos[t], u, fun Sin[t]}, u1,{t,0,2Pi}, ViewPoint -> {8,umax+1,umax+1} ] :[font = text; inactive; preserveAspect; endGroup; ] This defines the command Surface which will graph surfaces of revolution about the x-axis: ;[s] 3:0,0;24,1;33,2;91,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Courier,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Example :[font = text; inactive; Cclosed; preserveAspect; startGroup; ] Here is Example 4 on page 394: :[font = input; preserveAspect; endGroup; ] f[x_] := E^-x :[font = text; inactive; Cclosed; preserveAspect; startGroup; ] Its graph is: :[font = input; preserveAspect; endGroup; ] Plot[ f[x], {x,1,2} ]; :[font = text; inactive; Cclosed; preserveAspect; startGroup; ] Here is the surface obtained by revolving that curve about the x-axis: :[font = input; preserveAspect; ] Surface[ f[x], {x,1,2} ]; :[font = text; inactive; preserveAspect; endGroup; endGroup; ] Notice that the graph is really a collection of flat panels stitched together to form a patchwork quilt that closely matches the actual surface. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Exercises :[font = text; inactive; preserveAspect; ] For each of the following functions: (a) define f; (b) plot the graph of f over the interval indicated; (c) plot the surface obtained by revolving the graph of f about the x-axis; :[font = text; inactive; preserveAspect; ] 1. Exercise 9 on page 395. :[font = text; inactive; preserveAspect; ] 2. Exercise 10 on page 395. :[font = text; inactive; preserveAspect; ] 3. Exercise 15 on page 395. Use h=5 and a=2. (See Quiz 4.) :[font = text; inactive; preserveAspect; ] 4. A sphere of radius 5. :[font = text; inactive; preserveAspect; endGroup; endGroup; ] 5. Exercise 19 on page 395. To make give hole a radius of 3, use {x,-4,4} for the domain. :[font = section; inactive; Cclosed; pageBreak; preserveAspect; startGroup; ] The Volume of the Solid of Revolution :[font = text; inactive; Cclosed; preserveAspect; startGroup; ] The formula for the volume of a solid obtained from revolving the graph of a function f about the x-axis is the integral of p*f(x)^2 over the same interval: ;[s] 6:0,0;85,1;88,2;124,3;125,4;133,5;156,-1; 6:1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; ] volume = Integrate[ Pi*f[x]^2, {x,a,b} ] :[font = text; inactive; preserveAspect; endGroup; ] This formula is often called the "disk method." :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Example :[font = text; inactive; Cclosed; preserveAspect; startGroup; ] Continue now with our previous example (Example 3 on page 394): :[font = input; preserveAspect; ] f[x_] := E^-x :[font = input; preserveAspect; ] Integrate[ Pi*f[x]^2, {x,1,2} ] :[font = input; preserveAspect; ] Simplify[%] :[font = text; inactive; preserveAspect; endGroup; endGroup; ] This is the answer in the book. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Exercises :[font = text; inactive; preserveAspect; ] For the same functions given in the first exercise set, have Mathematica compute and simplify the volume of the solid obtained by revolving the curve about the x-axis: ;[s] 3:0,0;61,1;72,2;167,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = text; inactive; preserveAspect; ] 1. Exercise 9 on page 395. :[font = text; inactive; preserveAspect; ] 2. Exercise 10 on page 395. :[font = text; inactive; preserveAspect; ] 3. Exercise 15 on page 395. Use h=5 and a=2. :[font = text; inactive; preserveAspect; ] 4. A sphere of radius 5. :[font = text; inactive; preserveAspect; endGroup; endGroup; ] 5. Exercise 19 on page 395. To make give hole a radius of 3, use {x,-4,4} for the domain. :[font = section; inactive; Cclosed; pageBreak; preserveAspect; startGroup; ] Using More Than One Curve :[font = text; inactive; Cclosed; preserveAspect; startGroup; ] In this section, we'll need the external Mathematica package FilledPlot.m which is in the Graphics directory. To access this package, execute the following command: ;[s] 7:0,0;41,1;52,2;60,3;74,4;89,5;99,6;166,-1; 7:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; endGroup; ] <<Graphics/FilledPlot.m :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Examples :[font = text; inactive; Cclosed; preserveAspect; startGroup; ] The region R in Exercise 11 on page 395 is bounded between two curves: y=x^2 and y=x^3. For efficient analysis, first define f, g, a, and b: ;[s] 15:0,0;11,1;12,2;70,3;77,4;80,5;86,6;125,7;127,8;128,9;130,10;131,11;133,12;138,13;140,14;142,-1; 15:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; endGroup; ] f[x_] := x^2; g[x_] := x^3; a = 0; b = 1; :[font = text; inactive; Cclosed; preserveAspect; startGroup; ] Now use the FilledPlot command to graph the region R: ;[s] 3:0,0;51,1;52,2;54,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; endGroup; ] FilledPlot[ {f[x],g[x]}, {x,a,b} ]; :[font = text; inactive; Cclosed; preserveAspect; startGroup; ] The area of this region is: :[font = input; preserveAspect; endGroup; ] Integrate[ f[x]-g[x], {x,a,b} ] :[font = text; inactive; Cclosed; preserveAspect; startGroup; ] To graph the solid generated by revolving this region about the x-axis, we'll need two separate surfaces: :[font = input; preserveAspect; ] s1 = Surface[ f[x], {x,a,b} ]; :[font = input; preserveAspect; ] s2 = Surface[ g[x], {x,a,b} ]; :[font = text; inactive; preserveAspect; endGroup; ] These appear very similar. But we can tell from the FilledPlot above that the surface s2 must lie inside the surface s1 . ;[s] 6:0,0;52,1;64,2;86,3;90,4;117,5;124,-1; 6:1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0; :[font = text; inactive; Cclosed; preserveAspect; startGroup; ] The two surfaces together form the complete boundary of the solid: :[font = input; preserveAspect; endGroup; ] Show[ s1, s2 ]; :[font = text; inactive; Cclosed; preserveAspect; startGroup; ] Here's another view that makes it easier to see that there is some space between the inside and outside surfaces: :[font = input; preserveAspect; ] Show[ s1, s2, ViewPoint -> {8,7,1} ]; :[font = text; inactive; preserveAspect; endGroup; ] The ViewPoint option allows you to see the solid from different points of view. :[font = text; inactive; Cclosed; pageBreak; preserveAspect; startGroup; ] Now we can solve Exercise 11 by computing the volumes of the solids bounded by s1 and s2 separately and then taking their difference: ;[s] 5:0,0;78,1;82,2;85,3;89,4;1145,-1; 5:1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; ] v1 = Integrate[ Pi*f[x]^2, {x,a,b} ] :[font = input; preserveAspect; ] v2 = Integrate[ Pi*g[x]^2, {x,a,b} ] :[font = input; preserveAspect; ] v = v1 - v2 :[font = text; inactive; preserveAspect; endGroup; endGroup; ] That's called "making a difference..." :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Exercises :[font = text; inactive; preserveAspect; ] For each of the following: (a) define f, g, a, and b, ; (b) use the FilledPlot command to graph the given region R; (c) find the area of R; (d) define s1 and s2 to be the bounding surfaces for the solid of revolution; (e) show the resulting solid, from several viewpoints; (f) find the volume of the solid. ;[s] 19:0,0;38,1;40,2;41,3;43,4;44,5;46,6;51,7;53,8;69,9;81,10;115,11;116,12;140,13;141,14;154,15;158,16;161,17;165,18;312,-1; 19:1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = text; inactive; preserveAspect; ] 1. Exercise 12 on page 395. :[font = text; inactive; preserveAspect; ] 2. Exercise 13 on page 395. :[font = text; inactive; preserveAspect; ] 3. Exercise 14 on page 395. Use h=5 and a=2. :[font = text; inactive; preserveAspect; endGroup; endGroup; endGroup; ] 4. Let R be the circle bounded above by y=3+Sqrt[1-x^2] and below by y=3+Sqrt[1-x^2] . The resulting solid of revolution is a torus (like a doughnut). ;[s] 7:0,0;8,1;9,2;40,3;57,4;73,5;90,6;157,-1; 7:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; ^*) 
