9.2E: Exercises
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Exercises 9.2E.1−14
In the following exercises, specify whether the region is of Type I or Type II.
1. The region D bounded by y=x3, y=x3+1, x=0, and x=1 as given in the following figure.
2. Find the average value of the function f(x,y)=3xy on the region graphed in the previous exercise.
- Answer
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2720
3. Find the area of the region D given in the previous exercise.
4. The region D bounded by y=sin x, y=1+sin x, x=0, and x=π2 as given in the following figure.
- Answer
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Type I but not Type II
5. Find the average value of the function f(x,y)=cos x on the region graphed in the previous exercise.
6. Find the area of the region D given in the previous exercise.
- Answer
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π2
7. The region D bounded by x=y2−1 and x=√1−y2 as given in the following figure.
8. Find the volume of the solid under the graph of the function f(x,y)=xy+1 and above the region in the figure in the previous exercise.
- Answer
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16(8+3π)
9. The region D bounded by y=0, x=−10+y, and x=10−y as given in the following figure.
10. Find the volume of the solid under the graph of the function f(x,y)=x+y and above the region in the figure from the previous exercise.
- Answer
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10003
11. The region D bounded by y=0, x=y−1, x=π2 as given in the following figure.
12. The region D bounded by y=0 and y=x2−1 as given in the following figure.
- Answer
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Type I and Type II
13. Let D be the region bounded by the curves of equations y=x, y=−x and y=2−x2. Explain why D is neither of Type I nor II.
14. Let D be the region bounded by the curves of equations y=cos x and y=4−x2 and the x-axis. Explain why D is neither of Type I nor II.
- Answer
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The region D is not of Type I: it does not lie between two vertical lines and the graphs of two continuous functions g1(x) and g2(x). The region is not of Type II: it does not lie between two horizontal lines and the graphs of two continuous functions h1(y) and h2(y).
Exercises 9.2E.15−20
In the following exercises, evaluate the double integral ∬Df(x,y)dA over the region D.
15. f(x,y)=2x+4y and
D={(x,y)|0≤x≤1, x3≤y≤x3+1}
16. f(x,y)=1 and
D={(x,y)|0≤x≤π2, sinx≤y≤1+sinx}
- Answer
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π2
17. f(x,y)=2 and
D={(x,y)|0≤y≤1, y−1≤x≤arccosy}
18. f(x,y)=xy and
D={(x,y)|−1≤y≤1, y2−1≤x≤√1−y2}
- Answer
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0
19. f(x,y)=sin y and D is the triangular region with vertices (0,0), (0,3), and (3,0)
20. f(x,y)=−x+1 and D is the triangular region with vertices (0,0), (0,2), and (2,2)
- Answer
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23
Exercises 9.2E.21−26
Evaluate the iterated integrals.
21. ∫30∫3x2x(x+y2)dy dx
22. ∫10∫2√x+12√x(xy+1)dy dx
- Answer
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4120
23. ∫e2e∫2ln u(v+ln u)dv du
24. ∫21∫−u−u2−1(8uv)dv du
- Answer
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−63
25. ∫10∫√1−y2−√1−y2(2x+4y3)dx dy
26. ∫10∫√1−4y2−√1−4y24dx dy
- Answer
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π
Exercises 9.2E.27−29
27. Let D be the region bounded by y=1−x2, y=4−x2, and the x- and y-axes.
a. Show that
∬DxdA=∫10∫4−x21−x2x dy dx+∫21∫4−x20x dy dx by dividing the region D into two regions of Type I.
b. Evaluate the integral ∬Ds dA.
28. Let D be the region bounded by y=1, y=x, y=ln x, and the x-axis.
a. Show that
∬Dy2dA=∫0−1∫2−x2−xy2dy dx+∫10∫2−x2xy2dy dx by dividing the region D into two regions of Type I, where D=(x,y)|y≥x,y≥−x, y≤2−x2.
b. Evaluate the integral ∬Dy2dA.
29. Let D be the region bounded by y=x2, y=x+2, and y=−x.
a. Show that ∬Dx dA=∫10∫√y−yx dx dy+∫21∫√yy−2x dx dy by dividing the region D into two regions of Type II, where D=(x,y)|y≥x2, y≥−x, y≤x+2.
b. Evaluate the integral ∬Dx dA.
- Answer
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a. Answers may vary; b. 812
Exercises 9.2E.30−33
30. The region D bounded by x=0,y=x5+1, and y=3−x2 is shown in the following figure. Find the area A(D) of the region D.
31. The region D bounded by y=cos x, y=4 cos x, and x=±π3 is shown in the following figure. Find the area A(D) of the region D.
- Answer
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8π3
32. Find the area A(D) of the region D={(x,y)|y≥1−x2,y≤4−x2, y≥0, x≥0}.
33. Let D be the region bounded by y=1, y=x, y=ln x, and the x-axis. Find the area A(D) of the region D.
- Answer
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e−32
Exercises 9.2E.34−35
34. Find the average value of the function f(x,y)=sin y on the triangular region with vertices (0,0), (0,3), and (3,0).
35. Find the average value of the function f(x,y)=−x+1 on the triangular region with vertices (0,0), (0,2), and (2,2).
- Answer
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23
Exercises 9.2E.36−39
In the following exercises, change the order of integration and evaluate the integral.
36. ∫π/2−1∫x+10sin x dy dx
37. ∫10∫1−xx−1x dy dx
- Answer
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∫10∫1−xx−1x dy dx=∫0−1∫y+10x dx dy+∫10∫1−y−x dx dy=13
38. ∫0−1∫√y+1−√y+1y2dx dy
39. ∫1/2−1/2∫√y2+1−√y2+1y dx dy
- Answer
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∫1/2−1/2∫√y2+1−√y2+1y dx dy=∫21∫√x2−1−√x2−1y dy dx=0
Exercises 9.2E.40−41
40. The region D is shown in the following figure. Evaluate the double integral ∬D(x2+y)dA by using the easier order of integration.
41. The region D is shown in the following figure. Evaluate the double integral ∬D(x2−y2)dA by using the easier order of integration.
- Answer
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∬D(x2−y2)dA=∫1−1∫1−y4y4−1(x2−y2)dx dy=4644095
Exercises 9.2E.42−45
42. Find the volume of the solid under the surface z=2x+y2 and above the region bounded by y=x5 and y=x.
43. Find the volume of the solid under the plane z=3x+y and above the region determined by y=x7 and y=x.
- Answer
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45
44. Find the volume of the solid under the plane z=3x+y and above the region bounded by x=tan y, x=−tan y, and x=1.
45. Find the volume of the solid under the surface z=x3 and above the plane region bounded by x=sin y, x=−sin y, and x=1.
- Answer
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5π32
Exercises 9.2E.46−47
46. Let g be a positive, increasing, and differentiable function on the interval [a,b]. Show that the volume of the solid under the surface z=g′(x) and above the region bounded by y=0, y=g(x), x=a, and x=b is given by 12(g2(b)−g2(a)).
47. Let g be a positive, increasing, and differentiable function on the interval [a,b] and let k be a positive real number. Show that the volume of the solid under the surface z=g′(x) and above the region bounded by y=g(x), y=g(x)+k, x=a, and x=b is given by k(g(b)−g(a)).
Exercises 9.2E.48−51
48. Find the volume of the solid situated in the first octant and determined by the planes z=2, z=0, x+y=1, x=0, and y=0.
49. Find the volume of the solid situated in the first octant and bounded by the planes x+2y=1, x=0, z=4, and z=0.
- Answer
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1
50. Find the volume of the solid bounded by the planes x+y=1, x−y=1, x=0, z=0, and z=10.
51. Find the volume of the solid bounded by the planes x+y=1, x−y=1, x−y=−1, z=1, and z=0
- Answer
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2
Exercises 9.2E.52
52. Let S1 and S2 be the solids situated in the first octant under the planes x+y+z=1 and x+y+2z=1 respectively, and let S be the solid situated between S1, S2, x=0, and y=0.
- Find the volume of the solid S1.
- Find the volume of the solid S2.
- Find the volume of the solid S by subtracting the volumes of the solids S1 and S2.
53. Let S1 and S2 be the solids situated in the first octant under the planes 2x+2y+z=2 and x+y+z=1 respectively, and let S be the solid situated between S1, S2, x=0, and y=0.
- Find the volume of the solid S1.
- Find the volume of the solid S2.
- Find the volume of the solid S by subtracting the volumes of the solids S1 and S2.
- Answer
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a. 13; b. 16; c. 16
54. Let S1 and S2 be the solids situated in the first octant under the plane x+y+z=2 and under the sphere x2+y2+z2=4, respectively. If the volume of the solid S2 is 4π3 determine the volume of the solid S situated between S1 and S2 by subtracting the volumes of these solids.
55. Let S1 and S2 be the solids situated in the first octant under the plane x+y+z=2 and under the sphere x2+y2=4, respectively.
- Find the volume of the solid S1.
- Find the volume of the solid S2.
- Find the volume of the solid S situated between S1 and S2 by subtracting the volumes of the solids S1 and S2.
- Answer
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a. 43; b. 2π; c. 6π−43
Exercises 9.2E.56−57
56. [T] The following figure shows the region D bounded by the curves y=sin x, x=0, and y=x4. Use a graphing calculator or CAS to find the