ISM_T11_C15_B - - X X ~ X

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950 Chapter 15 Multiple Integrals Clear[x, y, f] f[x_, y_]:= 1 / (x y) Integrate[f[x, y], {x, 1, 3}, {y, 1, x}] To reverse the order of integration, it is best to first plot the region over which the integration extends. This can be done with ImplicitPlot and all bounds involving both x and y can be plotted. A graphics package must be loaded. Remember to use the double equal sign for the equations of the bounding curves. Clear[x, y, f] <<Graphics`ImplicitPlot` ImplicitPlot[{x==2y, x==4, y==0, y==1},{x, 0, 4.1}, {y, 0, 1.1}]; f[x_, y_]:=Exp[x ] 2 Integrate[f[x, y], {x, 0, 2}, {y, 0, x/2}] Integrate[f[x, y], {x, 2, 4}, {y, 0, 1}] ± To get a numerical value for the result, use the numerical integrator, . Verify that this equals the original. NIntegrate Integrate[f[x, y], {x, 0, 2}, {y, 0, x/2}] NIntegrate[f[x, y], {x, 2, 4}, {y, 0, 1}] ± NIntegrate[f[x, y], {y, 0, 1},{x, 2y, 4}] Another way to show a region is with the FilledPlot command. This assumes that functions are given as y = f(x). Clear[x, y, f] <<Graphics`FilledPlot` FilledPlot[{x , 9},{x, 0,3}, AxesLabels {x, y}]; 2 Ä f[x_, y_]:= x Cos[y ] 2 Integrate[f[x, y], {y, 0, 9}, {x, 0, Sqrt[y]}] 67. dy dx 0.603 68. e dy dx 0.558 '' 11 00 3x xy " xy ¸¸ ˆ‰ ## 69. tan xy dy dx 0.233 70. 3 1 x y dy dx 3.142 10 x ±" ¸² ² ¸ È # È 71. Evaluate the integrals: e dx dy 02 y 14 x # e dy dx œ± 20 2x / 2 41 xx e 2 erfi 2 2 erfi 4 œ² ± ² ± "" 44 4 ÈÈ ab 1.1494 10 ¸‚ 6 The following graphs was generated using Mathematica.
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Section 15.2 Areas, Moments, and Centers of Mass 951 72. Evaluate the integrals: x cos y dy dx x cos y dx dy '' 0x 00 39 9 y 22 2 ab œ È 0.157472 œ¸ ± sin 81 4 The following graphs was generated using Mathematica. 73. Evaluate the integrals: x y xy dx dy x y xy dy dx 0y / 3 2 24 2 y 8 x 32 3 È È ±œ ± 97.4315 67,520 693 The following graphs was generated using Mathematica. 74. Evaluate the integrals: e dx dy e dy dx y 4 4 x # xy xy œ È 20.5648 ¸ The following graphs was generated using Mathematica.
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952 Chapter 15 Multiple Integrals 75. Evaluate the integrals: dy dx '' 10 2x # 1 xy ± dx dy dx dy œ± 01 1 y 12 42 11 ±± È 1 ln 0.909543 ²± ¸ ˆ‰ 27 4 The following graphs was generated using Mathematica. 76. Evaluate the integrals: dx dy dy dx 1y 28 8 x 3 3 ÈÈ 22 œ È 0.866649 ¸ The following graphs was generated using Mathematica. 15.2 AREAS, MOMENTS, AND CENTERS OF MASS 1. dy dx (2 x) dx 2x 2, ' 00 0 x 2 œ²œ ² œ ’“ x 2 # # ! or dx dy (2 y) dy 2 ' 0 y 2 2. dy dx (4 2x) dx 4x x 4, ' 02 x 0 24 2 2 0 œ² œ ² œ cd # or dx dy dy 4 ' 0 4y 2 4 Î œœ y # 3. dx dy y y 2 dy ' 2y 2 2 1 # ² ± ab # 2y œ² ² ± yy 3 $# # " ²# 4 œ²² ± ² ²² œ ˆ "" ## 33 89
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Section 15.2 Areas, Moments, and Centers of Mass 953 4. dx dy 2y y dy y '' ' 0y 0 2y y 2 # œ±œ ± ab’“ ## # ! y 3 $ 4 84 33 5. dy dx e dx e 2 1 1 ' 00 0 ln 2 e ln 2 xx ln 2 0 x œ œ cd 6. dy dx ln x dx x ln x x ' 1l n x 1 e2 l n x e e 1 œœ ± (e e) (0 1) 1 œ±±±œ 7. dx dy 2y 2y dy y y ' 0 12 y y 1 # # œ± œ ± ab ± $ " ! 2 3 œ " 3 8. dx dy y 1 2y 2 dy ' y2 1 1y 1 1 # # ± ² 1 y œ ±œ ' 1 1 # " ±" y 4 $ 9. dx dy 2y dy y ' 3 0 62 y 6 # Î ± Š‹ ’“ yy 39 # $ # ' !
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ISM_T11_C15_B - - X X ~ X

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