ISM_T11_C15_B

# 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 ' ' ' ' 1 1 0 0 3 x 1 1 x y " xy ¸ ¸ 69. tan xy dy dx 0.233 70. 3 1 x y dy dx 3.142 ' ' ' ' 0 0 1 0 1 1 1 1 x " # # ¸ ¸ È 71. Evaluate the integrals: e dx dy ' ' 0 2y 1 4 x e dy dx e dy dx œ ' ' ' ' 0 0 2 0 2 x/2 4 1 x x e 2 erfi 2 2 erfi 4 œ " " 4 4 4 ˆ È È a b a b 1 1 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 ' ' ' ' 0 x 0 0 3 9 9 y 2 2 2 a b a b œ 0.157472 œ ¸ sin 81 4 a b The following graphs was generated using Mathematica. 73. Evaluate the integrals: x y xy dx dy x y xy dy dx ' ' ' ' 0 y 0 x /32 2 4 2y 8 x 2 2 2 2 3 2 3 a b a b œ 97.4315 œ ¸ 67,520 693 The following graphs was generated using Mathematica. 74. Evaluate the integrals: e dx dy e dy dx ' ' ' ' 0 0 0 0 2 4 y 4 4 x xy xy œ 20.5648 ¸ The following graphs was generated using Mathematica.
952 Chapter 15 Multiple Integrals 75. Evaluate the integrals: dy dx ' ' 1 0 2 x 1 x y dx dy dx dy œ ' ' ' ' 0 1 1 y 1 2 4 2 1 1 x y x y 1 ln 0.909543 ¸ ˆ 27 4 The following graphs was generated using Mathematica. 76. Evaluate the integrals: dx dy dy dx ' ' ' ' 1 y 1 1 2 8 8 x 3 3 1 1 x y x y È È 2 2 2 2 œ 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, ' ' ' 0 0 0 2 2 x 2 œ œ œ x 2 # ! or dx dy (2 y) dy 2 ' ' ' 0 0 0 2 2 y 2 œ œ 2. dy dx (4 2x) dx 4x x 4, ' ' ' 0 2x 0 2 4 2 2 0 œ œ œ c d # or dx dy dy 4 ' ' ' 0 0 0 4 y 2 4 œ œ y # 3. dx dy y y 2 dy ' ' ' 2 y 2 2 1 y 1 œ a b # 2y œ y y 3 # " # 2 2 4 œ œ ˆ ˆ " " # # 3 3 8 9

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Section 15.2 Areas, Moments, and Centers of Mass 953 4. dx dy 2y y dy y ' ' ' 0 y 0 2 y y 2 œ œ a b # # # ! y 3 4 œ œ 8 4 3 3 5. dy dx e dx e 2 1 1 ' ' ' 0 0 0 ln 2 e ln 2 x x ln 2 0 x œ œ œ œ c d 6. dy dx ln x dx x ln x x ' ' ' 1 ln x 1 e 2 ln x e e 1 œ œ c d (e e) (0 1) 1 œ œ 7. dx dy 2y 2y dy y y ' ' ' 0 y 0 1 2y y 1 œ œ a b # # \$ " ! 2 3 œ " 3 8. dx dy y 1 2y 2 dy ' ' ' 1 2y 2 1 1 y 1 1 œ a b # # 1 y dy y œ œ œ ' 1 1 a b # " " y 3 3 4 9. dx dy 2y dy y ' ' ' 0 y 3 0 6 2y 6 œ œ Š y y 3 9 # ' !
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• Spring '07
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