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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 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

\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

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

\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

\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

-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

\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

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.

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 \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.

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

Answer

\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

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

\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

\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

\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.

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 \iint_D (x^2 - y^2) 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

\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

\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

\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

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

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.

  1. Find the volume of the solid S_1.
  2. Find the volume of the solid S_2.
  3. 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.

  1. Find the volume of the solid S_1.
  2. Find the volume of the solid S_2.
  3. Find the volume of the solid S by subtracting the volumes of the solids S_1 and S_2.
Answer

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.

  1. Find the volume of the solid S_1.
  2. Find the volume of the solid S_2.
  3. 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


9.2E: Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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