9.2E: Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
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 ∬ over the region D.
15. f(x,y) = 2x + 4y and
D = \big\{(x,y)|\, 0 \leq x \leq 1, \space x^3 \leq y \leq x^3 + 1 \big\}
16. f(x,y) = 1 and
D = \big\{(x,y)| \, 0 \leq x \leq \frac{\pi}{2}, \space \sin x \leq y \leq 1 + \sin x \big\}
- Answer
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\frac{\pi}{2}
17. f(x,y) = 2 and
D = \big\{(x,y)| \, 0 \leq y \leq 1, \space y - 1 \leq x \leq \arccos y \big\}
18. f(x,y) = xy and
D = \big\{(x,y)| \, -1 \leq y \leq 1, \space y^2 - 1 \leq x \leq \sqrt{1 - y^2} \big\}
- Answer
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0
19. f(x,y) = sin \space y and D is the triangular region with vertices (0,0), \space (0,3), and (3,0)
20. f(x,y) = -x + 1 and D is the triangular region with vertices (0,0), \space (0,2), and (2,2)
- Answer
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\frac{2}{3}
Exercises \PageIndex{21-26}
Evaluate the iterated integrals.
21. \int_0^3 \int_{2x}^{3x} (x + y^2) dy \space dx
22. \int_0^1 \int_{2\sqrt{x}}^{2\sqrt{x}+1} (xy + 1) dy \space dx
- Answer
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\frac{41}{20}
23. \int_e^{e^2} \int_{ln \space u}^2 (v + ln \space u) dv \space du
24. \int_1^2 \int_{-u^2-1}^{-u} (8 uv) dv \space du
- Answer
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-63
25. \int_0^1 \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} (2x + 4y^3) dx \space dy
26. \int_0^1 \int_{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}} 4dx \space dy
- Answer
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\pi
Exercises \PageIndex{27-29}
27. Let D be the region bounded by y = 1 - x^2, \space y = 4 - x^2, and the x- and y-axes.
a. Show that
\iint_D xdA = \int_0^1 \int_{1-x^2}^{4-x^2} x \space dy \space dx + \int_1^2 \int_0^{4-x^2} x \space dy \space dx by dividing the region D into two regions of Type I.
b. Evaluate the integral \iint_D s \space dA.
28. Let D be the region bounded by y = 1, \space y = x, \space y = ln \space x, and the x-axis.
a. Show that
\iint_D y^2 dA = \int_{-1}^0 \int_{-x}^{2-x^2} y^2 dy \space dx + \int_0^1 \int_x^{2-x^2} y^2 dy \space dx by dividing the region D into two regions of Type I, where D = {(x,y)|y \geq x, y \geq -x, \space y \leq 2-x^2}.
b. Evaluate the integral \iint_D y^2 dA.
29. Let D be the region bounded by y = x^2, y = x + 2, and y = -x.
a. Show that \iint_D x \space dA = \int_0^1 \int_{-y}^{\sqrt{y}} x \space dx \space dy + \int_1^2 \int_{y-2}^{\sqrt{y}} x \space dx \space dy by dividing the region D into two regions of Type II, where D = {(x,y)|y \geq x^2, \space y \geq -x, \space y \leq x + 2}.
b. Evaluate the integral \iint_D x \space dA.
- Answer
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a. Answers may vary; b. \frac{8}{12}
Exercises \PageIndex{30-33}
30. The region D bounded by x = 0, y = x^5 + 1, and y = 3 - x^2 is shown in the following figure. Find the area A(D) of the region D.
31. The region D bounded by y = cos \space x, \space y = 4 \space cos \space x, and x = \pm \frac{\pi}{3} is shown in the following figure. Find the area A(D) of the region D.
- Answer
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\frac{8\pi}{3}
32. Find the area A(D) of the region D = \big\{(x,y)| \, y \geq 1 - x^2, y \leq 4 - x^2, \space y \geq 0, \space x \geq 0 \big\}.
33. Let D be the region bounded by y = 1, \space y = x, \space y = ln \space x, and the x-axis. Find the area A(D) of the region D.
- Answer
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e - \frac{3}{2}
Exercises \PageIndex{34-35}
34. Find the average value of the function f(x,y) = sin \space y on the triangular region with vertices (0,0), \space (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), \space (0,2), and (2,2).
- Answer
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\frac{2}{3}
Exercises \PageIndex{36-39}
In the following exercises, change the order of integration and evaluate the integral.
36. \int_{-1}^{\pi/2} \int_0^{x+1} sin \space x \space dy \space dx
37. \int_0^1 \int_{x-1}^{1-x} x \space dy \space dx
- Answer
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\int_0^1 \int_{x-1}^{1-x} x \space dy \space dx = \int_{-1}^0 \int_0^{y+1} x \space dx \space dy + \int_0^1 \int_-^{1-y} x \space dx \space dy = \frac{1}{3}
38. \int_{-1}^0 \int_{-\sqrt{y+1}}^{\sqrt{y+1}} y^2 dx \space dy
39. \int_{-1/2}^{1/2} \int_{-\sqrt{y^2+1}}^{\sqrt{y^2+1}} y \space dx \space dy
- Answer
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\int_{-1/2}^{1/2} \int_{-\sqrt{y^2+1}}^{\sqrt{y^2+1}} y \space dx \space dy = \int_1^2 \int_{-\sqrt{x^2-1}}^{\sqrt{x^2-1}} y \space dy \space dx = 0
Exercises \PageIndex{40-41}
40. The region D is shown in the following figure. Evaluate the double integral \iint_D (x^2 + y) dA by using the easier order of integration.
41. The region D is shown in the following figure. Evaluate the double integral \iint_D (x^2 - y^2) dA by using the easier order of integration.
- Answer
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\iint_D (x^2 - y^2) dA = \int_{-1}^1 \int_{y^4-1}^{1-y^4} (x^2 - y^2)dx \space dy = \frac{464}{4095}
Exercises \PageIndex{42-45}
42. Find the volume of the solid under the surface z = 2x + y^2 and above the region bounded by y = x^5 and y = x.
43. Find the volume of the solid under the plane z = 3x + y and above the region determined by y = x^7 and y = x.
- Answer
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\frac{4}{5}
44. Find the volume of the solid under the plane z = 3x + y and above the region bounded by x = tan \space y, \space x = -tan \space y, and x = 1.
45. Find the volume of the solid under the surface z = x^3 and above the plane region bounded by x = sin \space y, \space x = -sin \space y, and x = 1.
- Answer
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\frac{5\pi}{32}
Exercises \PageIndex{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, \space y = g(x), \space x = a, and x = b is given by \frac{1}{2}(g^2 (b) - g^2 (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), \space y = g(x) + k, \space x = a, and x = b is given by k(g(b) - g(a)).
Exercises \PageIndex{48-51}
48. Find the volume of the solid situated in the first octant and determined by the planes z = 2, z = 0, \space x + y = 1, \space 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, \space 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, \space x - y = 1, \space x = 0, \space z = 0, and z = 10.
51. Find the volume of the solid bounded by the planes x + y = 1, \space x - y = 1, \space x - y = -1, \space z = 1, and z = 0
- Answer
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2
Exercises \PageIndex{52}
52. Let S_1 and S_2 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 S_1, \space S_2, \space x = 0, and y = 0.
- Find the volume of the solid S_1.
- Find the volume of the solid S_2.
- Find the volume of the solid S by subtracting the volumes of the solids S_1 and S_2.
53. Let S_1 and S_2 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 S_1, \space S_2, \space x = 0, and y = 0.
- Find the volume of the solid S_1.
- Find the volume of the solid S_2.
- Find the volume of the solid S by subtracting the volumes of the solids S_1 and S_2.
- Answer
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a. \frac{1}{3}; b. \frac{1}{6}; c. \frac{1}{6}
54. Let S_1 and S_2 be the solids situated in the first octant under the plane x + y + z = 2 and under the sphere x^2 + y^2 + z^2 = 4, respectively. If the volume of the solid S_2 is \frac{4\pi}{3} determine the volume of the solid S situated between S_1 and S_2 by subtracting the volumes of these solids.
55. Let S_1 and S_2 be the solids situated in the first octant under the plane x + y + z = 2 and under the sphere x^2 + y^2 = 4, respectively.
- Find the volume of the solid S_1.
- Find the volume of the solid S_2.
- Find the volume of the solid S situated between S_1 and S_2 by subtracting the volumes of the solids S_1 and S_2.
- Answer
-
a. \frac{4}{3}; b. 2\pi; c. \frac{6\pi - 4}{3}
Exercises \PageIndex{56-57}
56. [T] The following figure shows the region D bounded by the curves y = sin \space x, \space x = 0, and y = x^4. Use a graphing calculator or CAS to find the