2.4E: Exercises

Exercises 1 - 4

Evaluate the triple integrals over the rectangular solid box $B$.

1. $\displaystyle \iiint_B (2x + 3y^2 + 4z^3) \, dV,$ where $B = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \, 0 \leq y \leq 2, \, 0 \leq z \leq 3\big\}$

Answer

$192$

2. $\displaystyle \iiint_B (xy + yz + xz) \, dV,$ where $B = \big\{(x,y,z) \,|\, 1 \leq x \leq 2, \, 0 \leq y \leq 2, \, 1 \leq z \leq 3\big\}$

3. $\displaystyle \iiint_B (x \cos y + z) \, dV,$ where $B = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \, 0 \leq y \leq \pi, \, -1 \leq z \leq 1\big\}$

Answer

$0$

4. $\displaystyle \iiint_B (z \sin x + y^2) \, dV,$ where $B = \big\{(x,y,z) \,|\, 0 \leq x \leq \pi, \, 0 \leq y \leq 1, \, -1 \leq z \leq 2\big\}$

Exercises 5 - 8

Change the order of integration by integrating first with respect to $z$, then $x$, then $y$.

5. $\displaystyle \int_0^1 \int_1^2 \int_2^3 (x^2 + \ln y + z) \, dx \, dy \, dz$

Answer

$\displaystyle \int_0^1 \int_1^2 \int_2^3 (x^2 + \ln y + z) \, dx \, dy \, dz = \frac{35}{6} + 2 \ln 2$

6. $\displaystyle \int_0^1 \int_{-1}^1 \int_0^3 (ze^x + 2y) \, dx \, dy \, dz$

7. $\displaystyle \int_{-1}^2 \int_1^3 \int_0^4 \left(x^2z + \frac{1}{y}\right) \, dx \, dy \, dz$

Answer

$\displaystyle \int_{-1}^2 \int_1^3 \int_0^4 \left(x^2z + \frac{1}{y}\right) \, dx \, dy \, dz = 64 + 12 \ln 3$

8. $\displaystyle \int_1^2 \int_{-2}^{-1} \int_0^1 \frac{x + y}{z} \, dx \, dy \, dz$

9. Let $F$, $G$, and $H$ be continuous functions on $[a,b]$, $[c,d]$, and $[e,f]$, respectively, where $a, \, b, \, c, \, d, \, e$, and $f$ are real numbers such that $a < b, \, c < d$, and $e < f$. Show that

$$\int_a^b \int_c^d \int_e^f F (x) \, G (y) \, H(z) \, dz \, dy \, dx = \left(\int_a^b F(x) \, dx \right) \left(\int_c^d G(y) \, dy \right) \left(\int_e^f H(z) \, dz \right).\nonumber$$

10. Let $F$, $G$, and $H$ be differential functions on $[a,b]$, $[c,d]$, and $[e,f]$, respectively, where $a, \, b, \, c, \, d, \, e$, and $f$ are real numbers such that $a < b, \, c < d$, and $e < f$. Show that

$$\int_a^b \int_c^d \int_e^f F' (x) \, G' (y) \, H'(z) \, dz \, dy \, dx = [F (b) - F (a)] \, [G(d) - G(c)] \, H(f) - H(e)].\nonumber$$

Exercises 11 - 14

Evaluate the triple integrals over the bounded region.

$E = \big\{(x,y,z) \,|\, a \leq x \leq b, \, h_1 (x) \leq y \leq h_2 (x), \, e \leq z \leq f \big\}.$

11. $\displaystyle \iiint_E (2x + 5y + 7z) \, dV,$ where $E = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \, 0 \leq y \leq -x + 1, \, 1 \leq z \leq 2\big\}$

Answer

$\frac{77}{12}$

12. $\displaystyle \iiint_E (y \ln x + z) \, dV,$ where $E = \big\{(x,y,z) \,|\, 1 \leq x \leq e, \, 0 \leq y \ln x, \, 0 \leq z \leq 1\big\}$

13. $\displaystyle \iiint_E (\sin x + \sin y) \, dV,$ where $E = \big\{(x,y,z) \,|\, 0 \leq x \leq \frac{\pi}{2}, \, -\cos x \leq y \cos x, \, -1 \leq z \leq 1 \big\}$

Answer

$2$

14. $\displaystyle \iiint_E (xy + yz + xz ) dV$ where $E = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \, -x^2 \leq y \leq x^2, \, 0 \leq z \leq 1 \big\}$

Exercises 15 - 18

Evaluate the triple integrals over the indicated bounded region $E$.

15. $\displaystyle \iiint_E (x + 2yz) \, dV,$ where $E = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \, 0 \leq y \leq x, \, 0 \leq z \leq 5 - x - y \big\}$

Answer

$\frac{430}{120}$

16. $\displaystyle \iiint_E (x^3 + y^3 + z^3) \, dV,$ where $E = \big\{(x,y,z) \,|\, 0 \leq x \leq 2, \, 0 \leq y \leq 2x, \, 0 \leq z \leq 4 - x - y \big\}$

17. $\displaystyle \iiint_E y \, dV,$ where $E = \big\{(x,y,z) \,|\, -1 \leq x \leq 1, \, -\sqrt{1 - x^2} \leq y \leq \sqrt{1 - x^2}, \, 0 \leq z \leq 1 - x^2 - y^2 \big\}$

Answer

$0$

18. $\displaystyle \iiint_E x \, dV,$ where $E = \big\{(x,y,z) \,|\, -2 \leq x \leq 2, \, -4\sqrt{1 - x^2} \leq y \leq \sqrt{4 - x^2}, \, 0 \leq z \leq 4 - x^2 - y^2 \big\}$

Exercises 19 - 22

Evaluate the triple integrals over the bounded region $E$ of the form.

$E = \big\{(x,y,z) \,|\, g_1 (y) \leq x \leq g_2(y), \, c \leq y \leq d, \, e \leq z \leq f \big\}$.

19. $\displaystyle \iiint_E x^2 \, dV,$ where $E = \big\{(x,y,z) \,|\, 1 - y^2 \leq x \leq y^2 - 1, \, -1 \leq y \leq 1, \, 1 \leq z \leq 2 \big\}$

Answer

$-\frac{64}{105}$

20. $\displaystyle \iiint_E (y + \sin x) \, dV,$ where $E = \big\{(x,y,z) \,|\, -y^4 \leq x \leq y^4, \, 0 \leq y \leq 2, \, 0 \leq z \leq 4\big\}$

21. $\displaystyle \iiint_E (x - yz) \, dV,$ where $E = \big\{(x,y,z) \,|\, -y^6 \leq x \leq \sqrt{y}, \, 0 \leq y \leq 1x, \, -1 \leq z \leq 1 \big\}$

Answer

$\frac{11}{26}$

22. $\displaystyle \iiint_E z \, dV,$ where $E = \big\{(x,y,z) \,|\, 2 - 2y \leq x \leq 2 + \sqrt{y}, \, 0 \leq y \leq 1x, \, 2 \leq z \leq 3 \big\}$

Exercises 23 - 26

Evaluate the triple integrals over the bounded region.

$E = \big\{(x,y,z) \,|\, g_1(y) \leq x \leq g_2(y), \, c \leq y \leq d, \, u_1(x,y) \leq z \leq u_2 (x,y) \big\}$

23. $\displaystyle \iiint_E z \, dV,$ where $E = \big\{(x,y,z) \,|\, -y \leq x \leq y, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1 - x^4 - y^4 \big\}$

Answer

$\frac{113}{450}$

24. $\displaystyle \iiint_E (xz + 1) \, dV,$ where $E = \big\{(x,y,z) \,|\, 0 \leq x \leq \sqrt{y}, \, 0 \leq y \leq 2, \, 0 \leq z \leq 1 - x^2 - y^2 \big\}$

25. $\displaystyle \iiint_E (x - z) \, dV,$ where $E = \big\{(x,y,z) \,|\, - \sqrt{1 - y^2} \leq x \leq y, \, 0 \leq y \leq \frac{1}{2}x, \, 0 \leq z \leq 1 - x^2 - y^2 \big\}$

Answer

$\frac{1}{160}(6 \sqrt{3} - 41)$

26. $\displaystyle \iiint_E (x + y) \, dV,$ where $E = \big\{(x,y,z) \,|\, 0 \leq x \leq \sqrt{1 - y^2}, \, 0 \leq y \leq 1x, \, 0 \leq z \leq 1 - x \big\}$

Exercises 27 - 30

Evaluate the triple integrals over the bounded region.

$E = \big\{(x,y,z) \,|\, (x,y) \in D, \, u_1 (x,y) x \leq z \leq u_2 (x,y) \big\}$, where $D$ is the projection of $E$ onto the $xy$-plane.

27. $\displaystyle \iint_D \left(\int_1^2 (x + y) \, dz \right) \, dA,$ where $D = \big\{(x,y) \,|\, x^2 + y^2 \leq 1\big\}$

Answer

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

28. $\displaystyle \iint_D \left(\int_1^3 x (z + 1)\, dz \right) \, dA,$ where $D = \big\{(x,y) \,|\, x^2 -y^2 \geq 1, \, x \leq \sqrt{5}\big\}$

29. $\displaystyle \iint_D \left(\int_0^{10-x-y} (x + 2z) \, dz \right) \, dA,$ where $D = \big\{(x,y) \,|\, y \geq 0, \, x \geq 0, \, x + y \leq 10\big\}$

Answer

$1250$

30. $\displaystyle \iint_D \left(\int_0^{4x^2+4y^2} y \, dz \right) \, dA,$ where $D = \big\{(x,y) \,|\, x^2 + y^2 \leq 4, \, y \geq 1, \, x \geq 0\big\}$

31. The solid $E$ bounded by $y^2 + z^2 = 9, \, z = 0$, and $x = 5$ is shown in the Figure $\PageIndex{1}$. Evaluate the integral $\displaystyle \iiint_E z \, dV$ by integrating first with respect to $z$, then $y$, and then $x$.

Figure $\PageIndex{1}$: The solid $E$ bounded above by $y^2 + z^2 = 9, bounded below by z = 0$, and $x = 5$.

Answer

$\displaystyle \int_0^5 \int_{-3}^3 \int_0^{\sqrt{9-y^2}} z \, dz \, dy \, dx = 90$

32. The solid $E$ bounded by $y = \sqrt{x}, \, x = 4, \, y = 0$, and $z = 1$ is given in the Figure $\PageIndex{2}$. Evaluate the integral $\displaystyle \iiint_E xyz \, dV$ by integrating first with respect to $x$, then $y$, and then $z$.

Figure $\PageIndex{2}$: The solid $E$ bounded by $y = \sqrt{x}, \, x = 4, \, y = 0$, and $z = 1$.

33. [T] The volume of a solid $E$ is given by the integral $\displaystyle \int_{-2}^0 \int_x^0 \int_0^{x^2+y^2} \, dz \, dy \, dx.$ Use a computer algebra system (CAS) to graph $E$ and find its volume. Round your answer to two decimal places.

Answer

$V \approx 5.33 \text{ units}^3$

Figure $\PageIndex{3}$: The graph illustrates the solid $E$ bounded by $z = x^2 + y^2$, $z = 0$, $y = x$, $y = 0$, $x = 0$, and $x = 2$.

34. [T] The volume of a solid $E$ is given by the integral $\displaystyle \int_{-1}^0 \int_{-x^3}^0 \int_0^{1+\sqrt{x^2+y^2}} \, dz \, dy \, dx.$ Use a CAS to graph $E$ and find its volume $V$. Round your answer to two decimal places.

Exercises 35 - 38

Use two circular permutations of the variables $x$, $y$, and $z$ to write new integrals whose values equal the value of the original integral. A circular permutation of $x, \, y$, and $z$ is the arrangement of the numbers in one of the following orders: $y$, $z$, and $x$ or $z$, $x$, and $y$.

35. $\displaystyle \int_0^1 \int_1^3 \int_2^4 (x^2z^2 + 1) \, dx \, dy \, dz$

Answer

$\displaystyle \int_0^1 \int_1^3 \int_2^4 (y^2z^2 + 1) \, dz \, dx \, dy;$
$\displaystyle \int_0^1 \int_1^3 \int_2^4 (x^2y^2 + 1) \, dy \, dz \, dx$

36. $\displaystyle \int_0^3 \int_0^1 \int_0^{-x+1} (2x + 5y + 7z) dy \, dx \, dz$

37. $\displaystyle \int_0^1 \int_{-y}^y \int_0^{1-x^4-y^4} \ln x \, dz \, dx \, dy$

38. $\displaystyle \int_{-1}^1 \int_0^1 \int_{-y^6}^{\sqrt{y}} (x + yz) \, dx \, dy \, dz$

39. Set up the integral that gives the volume of the solid $E$ bounded by $y^2 = x^2 + z^2$ and $y = a^2$, where $a > 0$.

Answer

$\displaystyle V = \int_{-a}^a \int_{-\sqrt{a^2-z^2}}^{\sqrt{a^2-z^2}} \int_{\sqrt{x^2+z^2}}^{a^2} \, dy \, dx \, dz$

40. Set up the integral that gives the volume of the solid $E$ bounded by $x = y^2 + z^2$ and $x = a^2$, where $a > 0$.

Average Value of a Function

41. Find the average value of the function $f(x,y,z) = x + y + z$ over the parallelepiped determined by $x + 0, \, x = 1, \, y = 0, \, y = 3, \, z = 0$, and $z = 5$.

Answer

$\frac{9}{2}$

42. Find the average value of the function $f(x,y,z) = xyz$ over the solid $E = [0,1] \times [0,1] \times [0,1]$ situated in the first octant.

Finding Volumes using Triple Integrals

43. Find the volume of the solid $E$ that lies under the plane $x + y + z = 9$ and whose projection onto the $xy$-plane is bounded by $x = \sqrt{y-1}, \, x = 0$, and $x + y = 7$.

Answer

$\frac{156}{5} \text{ units}^3$

44. Find the volume of the solid $E$ that lies under the plane $2x+y+z=8$ and whose projection onto the $xy$-plane is bounded by $x = sin^{-1} y, \, y = 0$, and $x = \frac{\pi}{2}$.

45. Consider the pyramid with the base in the $xy$-plane of $[-2,2] \times [-2,2]$ and the vertex at the point $(0,0,8)$.

a. Show that the equations of the planes of the lateral faces of the pyramid are $4y + z = 8, \, 4y - z = -8, \, 4x + z = 8$, and $-4x + z = 8$.

b. Find the volume of the pyramid.

Answer

a. Answers may vary;
b. $\frac{128}{3} \text{ units}^3$

46. Consider the pyramid with the base in the $xy$-plane of $[-3,3] \times [-3,3]$ and the vertex at the point $(0,0,9)$.

a. Show that the equations of the planes of the side faces of the pyramid are $3y + z = 9, \, 3y + z = 9, \, y = 0$ and $x = 0$.

b. Find the volume of the pyramid.

47. The solid $E$ bounded by the sphere of equation $x^2 + y^2 + z^2 = r^2$ with $r > 0$ and located in the first octant is represented in the Figure $\PageIndex{4}$.

Figure $\PageIndex{4}$: The graph illustrates the annulus.

a. Write the triple integral that gives the volume of $E$ by integrating first with respect to $z$, then with $y$, and then with $x$.

b. Rewrite the integral in part a. as an equivalent integral in five other orders.

Answer

a. $\displaystyle \int_0^4 \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} \, dz \, dy \, dx$
b. $\displaystyle \int_0^2 \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} \, dz \, dx \, dy,$
$\displaystyle \int_0^r \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} \, dy \, dx \, dz,$
$\displaystyle \int_0^r \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} dy \, dz \, dx,$
$\displaystyle \int_0^r \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} dx \, dy \, dz,$
$\displaystyle \int_0^r \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} dx \, dz \,dy$

48. The solid $E$ bounded by the sphere of equation $9x^2 + 4y^2 + z^2 = 1$ and located in the first octant is represented in the Figure $\PageIndex{5}$.

Figure $\PageIndex{5}$: The graph illustrates the solid $E$ ounded by the sphere of equation $9x^2 + 4y^2 + z^2 = 1$ and located in the first octant.

a. Write the triple integral that gives the volume of $E$ by integrating first with respect to $z$ then with $y$ and then with $x$.

b. Rewrite the integral in part a. as an equivalent integral in five other orders.

49. Find the volume of the prism with vertices $(0,0,0), \, (2,0,0), \, (2,3,0), \, (0,3,0), \, (0,0,1)$, and $(2,0,1)$.

Answer

$3 \text{ units}^3$

50. Find the volume of the prism with vertices $(0,0,0), \, (4,0,0), \, (4,6,0), \, (0,6,0), \, (0,0,1)$, and $(4,0,1)$.

51. The solid $E$ bounded by $z = 10 - 2x - y$ and situated in the first octant is given in the Figure $\PageIndex{6}$. Find the volume of the solid.

Figure $\PageIndex{6}$: The graph illustrates the solid $E$ bounded by $z = 10 - 2x - y$ and situated in the first octant.

Answer

$\frac{250}{3} \text{ units}^3$

52. The solid $E$ bounded by $z = 1 - x^2$ and situated in the first octant is given in the Figure $\PageIndex{7}$. Find the volume of the solid.

Figure $\PageIndex{7}$: The graph illustrates the solid $E$ bounded by $z = 1 - x^2$ and situated in the first octant.

Approximating Triple Integrals

53. The midpoint rule for the triple integral $\displaystyle \iiint_B f(x,y,z) \,dV$ over the rectangular solid box $B$ is a generalization of the midpoint rule for double integrals. The region $B$ is divided into subboxes of equal sizes and the integral is approximated by the triple Riemann sum $$\sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f(\bar{x_i}, \bar{y_j}, \bar{z_k}) \Delta V,\nonumber$$ where $(\bar{x_i}, \bar{y_j}, \bar{z_k})$ is the center of the box $B_{ijk}$ and $\Delta V$ is the volume of each subbox. Apply the midpoint rule to approximate $$\iiint_B x^2 \,dV\nonumber$$ over the solid $B = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1 \big\}$ by using a partition of eight cubes of equal size. Round your answer to three decimal places.

Answer

$\displaystyle \iiint_B f(x,y,z) \,dV\quad$ $\approx\quad\frac{5}{16} \approx 0.313 \text{ units}^3$

54. [T]

a. Apply the midpoint rule to approximate $\displaystyle \iiint_B e^{-x^2} \, dV$ over the solid $B = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1 \big\}$ by using a partition of eight cubes of equal size. Round your answer to three decimal places.

b. Use a CAS to improve the above integral approximation in the case of a partition of $n^3$ cubes of equal size, where $n = 3,\,4,\, ..., \,10$.

Applications

55. Suppose that the temperature in degrees Celsius at a point $(x,y,z)$ of a solid $E$ bounded by the coordinate planes and the plane $x + y + z = 5$ is given by: $$T (x,y,z) = xz + 5z + 10\nonumber$$ Find the average temperature over the solid.

Answer

$17.5^{\circ}$ C

56. Suppose that the temperature in degrees Fahrenheit at a point $(x,y,z)$ of a solid $E$ bounded by the coordinate planes and the plane $x + y + z = 5$ is given by: $$T(x,y,z) = x + y + xy\nonumber$$ Find the average temperature over the solid.

57. Show that the volume of a right square pyramid of height $h$ and side length $a$ is $v = \dfrac{ha^2}{3}$ by using triple integrals.

58. Show that the volume of a regular right hexagonal prism of edge length $a$ is $\dfrac{3a^3 \sqrt{3}}{2}$ by using triple integrals.

59. Show that the volume of a regular right hexagonal pyramid of edge length $a$ is $\dfrac{a^3 \sqrt{3}}{2}$ by using triple integrals.

60. If the charge density at an arbitrary point $(x,y,z)$ of a solid $E$ is given by the function $\rho (x,y,z)$, then the total charge inside the solid is defined as the triple integral $\displaystyle \iiint_E \rho (x,y,z) \,dV.$ Assume that the charge density of the solid $E$ enclosed by the paraboloids $x = 5 - y^2 - z^2$ and $x = y^2 + z^2 - 5$ is equal to the distance from an arbitrary point of $E$ to the origin. Set up the integral that gives the total charge inside the solid $E$.

Answer

Total Charge inside the Solid $E \quad=\quad$ $\displaystyle \int_{-\sqrt{5}}^{\sqrt{5}}\int_{-\sqrt{5-y^2}}^{\sqrt{5-y^2}}\int_{y^2+z^2-5}^{5 - y^2 - z^2} \sqrt{x^2+y^2+z^2}\,dx\,dz\,dy$