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9.2E: Exercises

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Exercises 9.2E.114

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.

A region is bounded by y = 1 + x cubed, y = x cubed, x = 0, and x = 1.

2. Find the average value of the function f(x,y)=3xy on the region graphed in the previous exercise.

Answer

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.

A region is bounded by y = 1 + sin x, y = sin x, x = 0, and x = pi/2.

Answer

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

π2

7. The region D bounded by x=y21 and x=1y2 as given in the following figure.

A region is bounded by x = negative 1 + y squared and x = the square root of the quantity (1 minus y squared).

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

16(8+3π)

9. The region D bounded by y=0, x=10+y, and x=10y as given in the following figure.

A region is bounded by x = negative 10 + y, x = 10 minus y, and y = 0.

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

10003

11. The region D bounded by y=0, x=y1, x=π2 as given in the following figure.

A region is bounded by x = pi/2, y = 0, and x = negative 1 + y.

12. The region D bounded by y=0 and y=x21 as given in the following figure.

A region is bounded by y = 0 and y = negative 1 + x squared.

Answer

Type I and Type II

13. Let D be the region bounded by the curves of equations y=x, y=x and y=2x2. 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=4x2 and the x-axis. Explain why D is neither of Type I nor II.

Answer

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.1520

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)|0x1, x3yx3+1}

16. f(x,y)=1 and

D={(x,y)|0xπ2, sinxy1+sinx}

Answer

π2

17. f(x,y)=2 and

D={(x,y)|0y1, y1xarccosy}

18. f(x,y)=xy and

D={(x,y)|1y1, y21x1y2}

Answer

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

23

Exercises 9.2E.2126

Evaluate the iterated integrals.

21. 303x2x(x+y2)dy dx

22. 102x+12x(xy+1)dy dx

Answer

4120

23. e2e2ln u(v+ln u)dv du

24. 21uu21(8uv)dv du

Answer

63

25. 101y21y2(2x+4y3)dx dy

26. 1014y214y24dx dy

Answer

π

Exercises 9.2E.2729

27. Let D be the region bounded by y=1x2, y=4x2, and the x- and y-axes.

a. Show that

DxdA=104x21x2x dy dx+214x20x 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=012x2xy2dy dx+102x2xy2dy dx by dividing the region D into two regions of Type I, where D=(x,y)|yx,yx, y2x2.

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=10yyx dx dy+21yy2x dx dy by dividing the region D into two regions of Type II, where D=(x,y)|yx2, yx, yx+2.

b. Evaluate the integral Dx dA.

Answer

a. Answers may vary; b. 812

Exercises 9.2E.3033

30. The region D bounded by x=0,y=x5+1, and y=3x2 is shown in the following figure. Find the area A(D) of the region D.

A region is bounded by y = 1 + x to the fifth power, y = 3 minus x squared, and x = 0.

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.

A region is bounded by y = cos x, y = 4 + cos x, x = negative 1, and x = 1.

Answer

8π3

32. Find the area A(D) of the region D={(x,y)|y1x2,y4x2, y0, x0}.

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

e32

Exercises 9.2E.3435

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

23

Exercises 9.2E.3639

In the following exercises, change the order of integration and evaluate the integral.

36. π/21x+10sin x dy dx

37. 101xx1x dy dx

Answer

101xx1x dy dx=01y+10x dx dy+101yx dx dy=13

38. 01y+1y+1y2dx dy

39. 1/21/2y2+1y2+1y dx dy

Answer

1/21/2y2+1y2+1y dx dy=21x21x21y dy dx=0

Exercises 9.2E.4041

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.

A region is bounded by y = negative 4 + x squared and y = 4 minus x squared.

41. The region D is shown in the following figure. Evaluate the double integral D(x2y2)dA by using the easier order of integration.

A region is bounded by y to the fourth power = 1 minus x and y to the fourth power = 1 + x.

Answer

D(x2y2)dA=111y4y41(x2y2)dx dy=4644095

Exercises 9.2E.4245

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

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

5π32

Exercises 9.2E.4647

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.4851

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

1

50. Find the volume of the solid bounded by the planes x+y=1, xy=1, x=0, z=0, and z=10.

51. Find the volume of the solid bounded by the planes x+y=1, xy=1, xy=1, z=1, and z=0

Answer

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.

  1. Find the volume of the solid S1.
  2. Find the volume of the solid S2.
  3. 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.

  1. Find the volume of the solid S1.
  2. Find the volume of the solid S2.
  3. Find the volume of the solid S by subtracting the volumes of the solids S1 and S2.
Answer

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.

  1. Find the volume of the solid S1.
  2. Find the volume of the solid S2.
  3. Find the volume of the solid S situated between S1 and S2 by subtracting the volumes of the solids S1 and S2.
Answer

a. 43; b. 2π; c. 6π43

Exercises 9.2E.5657

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


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