(*^ ::[paletteColors = 128; automaticGrouping; 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, 14, "Times"; ; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; 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;] :[font = title; inactive; preserveAspect; startGroup; ] Tank Problem :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Definite Integrals with Mathematica :[font = input; preserveAspect; ] f[x_] := x^3; a = 5; b = 10; Integrate[f[x], {x, a, b}] :[font = input; preserveAspect; endGroup; ] Integrate[x^3, {x, 5, 10}] :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Exercise 5 :[font = input; preserveAspect; startGroup; ] 4 r Integrate[Sqrt[1 - (x/r)^2], {x, 0, r}] :[font = output; output; inactive; preserveAspect; endGroup; ] Pi*r^2 ;[o] 2 Pi r :[font = input; preserveAspect; startGroup; ] 4 b Integrate[Sqrt[1 - (x/a)^2], {x, 0, a}] :[font = output; output; inactive; preserveAspect; endGroup; ] a*b*Pi ;[o] a b Pi :[font = input; preserveAspect; startGroup; ] 4 a Integrate[Sqrt[1 - (y/b)^2], {y, 0, b}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; ] a*b*Pi ;[o] a b Pi :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Exercise 6 :[font = input; preserveAspect; startGroup; ] 2 L a Integrate[Sqrt[1 - (y/b)^2], {y, -b, h}] :[font = output; output; inactive; preserveAspect; endGroup; ] 2*a*L*((h*(1 - h^2/b^2)^(1/2))/2 + (b*Pi)/4 + (b*ArcSin[h/b])/2) ;[o] 2 h h Sqrt[1 - --] h 2 b ArcSin[-] b b Pi b 2 a L (-------------- + ---- + -----------) 2 4 2 :[font = input; preserveAspect; startGroup; ] % //TeXForm :[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup; endGroup; ] 2\,a\,L\,\left( {{h\, {\sqrt{1 - {{{h^2}}\over {{b^2}}}}}}\over 2} + {{b\,\pi }\over 4} + {{b\,\arcsin ({h\over b})}\over 2} \right) ;[o] 2\,a\,L\,\left( {{h\,{\sqrt{1 - {{{h^2}}\over {{b^2}}}}}}\over 2} + {{b\,\pi }\over 4} + {{b\,\arcsin ({h\over b})}\over 2} \right) :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Exercise 7 :[font = input; preserveAspect; startGroup; ] L = 20; a = 10; b = 5; V[h_] = 2 L a Integrate[Sqrt[1 - (y/b)^2], {y, -b, h}]// Simplify :[font = output; output; inactive; preserveAspect; endGroup; ] 20*(2*h*(25 - h^2)^(1/2) + 25*Pi + 50*ArcSin[h/5]) ;[o] 2 h 20 (2 h Sqrt[25 - h ] + 25 Pi + 50 ArcSin[-]) 5 :[font = input; preserveAspect; startGroup; ] Table[{h, V[h]}, {h, -5, 5}] //N//TableForm :[font = output; output; inactive; preserveAspect; endGroup; endGroup; ] TableForm[{{-5., 0}, {-4., 163.5011087932844}, {-3., 447.2952180016123}, {-2., 792.6734251309414}, {-1., 1173.479226581911}, {0, 1570.796326794897}, {1., 1968.113427007882}, {2., 2348.919228458852}, {3., 2694.297435588182}, {4., 2978.091544796509}, {5., 3141.592653589793}}] ;[o] -5. 0 -4. 163.501 -3. 447.295 -2. 792.673 -1. 1173.48 0 1570.8 1. 1968.11 2. 2348.92 3. 2694.3 4. 2978.09 5. 3141.59 :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Exercise 8 :[font = input; preserveAspect; ] :[font = input; preserveAspect; startGroup; ] L = 20; a = 10; b = 5; V[h_] = 2 L a Integrate[Sqrt[1 - (y/b)^2], {y, -b, h}]//N Vprime[h_] = 2 L a Sqrt[1 - (h/b)^2]//N :[font = output; output; inactive; preserveAspect; ] 400.*(3.926990816987241 + 0.1*(h*(25. - 1.*h^2)^(1/2) + 25.*ArcSin[0.2*h])) ;[o] 2 400. (3.92699 + 0.1 (h Sqrt[25. - 1. h ] + 25. ArcSin[0.2 h])) :[font = output; output; inactive; preserveAspect; endGroup; ] 400.*(1. - 0.04*h^2)^(1/2) ;[o] 2 400. Sqrt[1. - 0.04 h ] :[font = input; preserveAspect; startGroup; ] c = 500.0; hn = -2.8; Do [ hn = hn - (V[hn] - c)/Vprime[hn]; Print[hn], {5}] :[font = print; inactive; preserveAspect; endGroup; ] -2.83758 -2.8377 -2.8377 -2.8377 -2.8377 :[font = input; preserveAspect; startGroup; ] c = 1000.0; hn = -1.5; Do [ hn = hn - (V[hn] - c)/Vprime[hn]; Print[hn], {5}] :[font = print; inactive; preserveAspect; endGroup; ] -1.44738 -1.44747 -1.44747 -1.44747 -1.44747 :[font = input; preserveAspect; startGroup; ] c = 1500.0; hn = -0.2; Do [ hn = hn - (V[hn] - c)/Vprime[hn]; Print[hn], {5}] :[font = print; inactive; preserveAspect; endGroup; endGroup; ] -0.177026 -0.177028 -0.177028 -0.177028 -0.177028 :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Exercise 9 :[font = input; preserveAspect; ] L = 20; a = 10; b = 5; Pi a b L //N :[font = input; preserveAspect; endGroup; endGroup; ] % / 2 ^*)