
# 7.2E: Exercises


## Exercises $$\PageIndex{1-14}$$

In the following exercises, specify whether the region is of Type I or Type II.

1. The region $$D$$ bounded by $$y = x^3, \space y = x^3 + 1, \space x = 0,$$ 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.

$$\frac{27}{20}$$

3. Find the area of the region $$D$$ given in the previous exercise.

4. The region $$D$$ bounded by $$y = sin \space x, \space y = 1 + sin \space x, \space x = 0$$, and $$x = \frac{\pi}{2}$$ as given in the following figure.

Type I but not Type II

5. Find the average value of the function $$f(x,y) = cos \space x$$ on the region graphed in the previous exercise.

6. Find the area of the region $$D$$ given in the previous exercise.

$$\frac{\pi}{2}$$

7. The region $$D$$ bounded by $$x = y^2 - 1$$ and $$x = \sqrt{1 - y^2}$$ 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.

$$\frac{1}{6}(8 + 3\pi)$$

9. The region $$D$$ bounded by $$y = 0, \space x = -10 + y,$$ 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.

$$\frac{1000}{3}$$

11. The region $$D$$ bounded by $$y = 0, \space x = y - 1, \space x = \frac{\pi}{2}$$ as given in the following figure.

12. The region $$D$$ bounded by $$y = 0$$ and $$y = x^2 - 1$$ as given in the following figure.

Type I and Type II

13. Let $$D$$ be the region bounded by the curves of equations $$y = x, \space y = -x$$ and $$y = 2 - x^2$$. Explain why $$D$$ is neither of Type I nor II.

14. Let $$D$$ be the region bounded by the curves of equations $$y = cos \space x$$ and $$y = 4 - x^2$$ and the $$x$$-axis. Explain why $$D$$ is neither of Type I nor II.

The region $$D$$ is not of Type I: it does not lie between two vertical lines and the graphs of two continuous functions $$g_1(x)$$ and $$g_2(x)$$. The region is not of Type II: it does not lie between two horizontal lines and the graphs of two continuous functions $$h_1(y)$$ and $$h_2(y)$$.

## Exercises $$\PageIndex{15-20}$$

In the following exercises, evaluate the double integral $$\iint_D f(x,y) dA$$ 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

$$D = \big\{(x,y)| \, 0 \leq x \leq \frac{\pi}{2}, \space \sin x \leq y \leq 1 + \sin x \big\}$$

$$\frac{\pi}{2}$$

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

$$D = \big\{(x,y)| \, -1 \leq y \leq 1, \space y^2 - 1 \leq x \leq \sqrt{1 - y^2} \big\}$$

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 $$(2,2)$$

$$\frac{2}{3}$$

## Exercises $$\PageIndex{21-26}$$

Evaluate the iterated integrals.

21. $\int_0^3 \int_{2x}^{3x} (x + y^2) dy \space dx$

22. $\int_0^1 \int_{2\sqrt{x}}^{2\sqrt{x}+1} (xy + 1) dy \space dx$

$$\frac{41}{20}$$

23. $\int_e^{e^2} \int_{ln \space u}^2 (v + ln \space u) dv \space du$

24. $\int_1^2 \int_{-u^2-1}^{-u} (8 uv) dv \space du$

$$-63$$

25. $\int_0^1 \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} (2x + 4y^3) dx \space dy$

26. $\int_0^1 \int_{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}} 4dx \space dy$

$$\pi$$

## 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 $\iint_D x \space dA.$

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 $$D$$.

$$\frac{8\pi}{3}$$

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 $$D$$.

$$e - \frac{3}{2}$$

## 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 $$(2,2)$$.

$$\frac{2}{3}$$

## 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. $\int_0^1 \int_{x-1}^{1-x} x \space dy \space dx$

$\int_0^1 \int_{x-1}^{1-x} x \space dy \space dx = \int_{-1}^0 \int_0^{y+1} x \space dx \space dy + \int_0^1 \int_-^{1-y} x \space dx \space dy = \frac{1}{3}$

38. $\int_{-1}^0 \int_{-\sqrt{y+1}}^{\sqrt{y+1}} y^2 dx \space dy$

39. $\int_{-1/2}^{1/2} \int_{-\sqrt{y^2+1}}^{\sqrt{y^2+1}} y \space dx \space dy$

$\int_{-1/2}^{1/2} \int_{-\sqrt{y^2+1}}^{\sqrt{y^2+1}} y \space dx \space dy = \int_1^2 \int_{-\sqrt{x^2-1}}^{\sqrt{x^2-1}} y \space dy \space dx = 0$

## 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.

$\iint_D (x^2 - y^2) dA = \int_{-1}^1 \int_{y^4-1}^{1-y^4} (x^2 - y^2)dx \space dy = \frac{464}{4095}$

## 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 $$y = x$$.

$$\frac{4}{5}$$

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 $$x = 1$$.

$$\frac{5\pi}{32}$$

## 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 $$z = 0$$.

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 $$z = 0$$

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 $$S_2$$.

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 $$S_2$$.
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