
# 14.5E: Exercises for Section 14.5


In exercises 1 - 8, evaluate the triple integrals $$\displaystyle \iiint_E f(x,y,z) \, dV$$ over the solid $$E$$.

1. $$f(x,y,z) = z, \quad B = \big\{(x,y,z)\, | \,x^2 + y^2 \leq 9, \quad x \leq 0, \quad y \leq 0, \quad 0 \leq z \leq 1\big\}$$

$$\frac{9\pi}{8}$$

2. $$f(x,y,z) = xz^2, \space B = \big\{(x,y,z)\, | \,x^2 + y^2 \leq 16, \space x \geq 0, \space y \leq 0, \space -1 \leq z \leq 1\big\}$$

3. $$f(x,y,z) = xy, \space B = \big\{(x,y,z)\, | \,x^2 + y^2 \leq 1, \space x \geq 0, \space x \geq y, \space -1 \leq z \leq 1\big\}$$

$$\frac{1}{8}$$

4. $$f(x,y,z) = x^2 + y^2, \space B = \big\{(x,y,z)\, | \,x^2 + y^2 \leq 4, \space x \geq 0, \space x \leq y, \space 0 \leq z \leq 3\big\}$$

5. $$f(x,y,z) = e^{\sqrt{x^2+y^2}}, \space B = \big\{(x,y,z)\, | \,1 \leq x^2 + y^2 \leq 4, \space y \leq 0, \space x \leq y\sqrt{3}, \space 2 \leq z \leq 3 \big\}$$

$$\frac{\pi e^2}{6}$$

6. $$f(x,y,z) = \sqrt{x^2 + y^2}, \space B = \big\{(x,y,z)\, | \,1 \leq x^2 + y^2 \leq 9, \space y \leq 0, \space 0 \leq z \leq 1\big\}$$

7. a. Let $$B$$ be a cylindrical shell with inner radius $$a$$ outer radius $$b$$, and height $$c$$ where $$0 < a < b$$ and $$c>0$$. Assume that a function $$F$$ defined on $$B$$ can be expressed in cylindrical coordinates as $$F(x,y,z) = f(r) + h(z)$$, where $$f$$ and $$h$$ are differentiable functions. If $$\displaystyle \int_a^b \bar{f} (r) \,dr = 0$$ and $$\bar{h}(0) = 0$$, where $$\bar{f}$$ and $$\bar{h}$$ are antiderivatives of $$f$$ and $$h$$, respectively, show that $$\displaystyle \iiint_B F(x,y,z) \,dV = 2\pi c (b\bar{f} (b) - a \bar{f}(a)) + \pi(b^2 - a^2) \bar{h} (c).$$

b. Use the previous result to show that $$\displaystyle \iiint_B \left(z + \sin \sqrt{x^2 + y^2}\right) \,dx \space dy \space dz = 6 \pi^2 ( \pi - 2),$$ where $$B$$ is a cylindrical shell with inner radius $$\pi$$ outer radius $$2\pi$$, and height $$2$$.

8. a. Let $$B$$ be a cylindrical shell with inner radius $$a$$ outer radius $$b$$ and height $$c$$ where $$0 < a < b$$ and $$c > 0$$. Assume that a function $$F$$ defined on $$B$$ can be expressed in cylindrical coordinates as $$F(x,y,z) = f(r) g(\theta) f(z)$$, where $$f, \space g,$$ and $$h$$ are differentiable functions. If $$\displaystyle\int_a^b \tilde{f} (r) \, dr = 0,$$ where $$\tilde{f}$$ is an antiderivative of $$f$$, show that $$\displaystyle\iiint_B F (x,y,z)\,dV = [b\tilde{f}(b) - a\tilde{f}(a)] [\tilde{g}(2\pi) - \tilde{g}(0)] [\tilde{h}(c) - \tilde{h}(0)],$$ where $$\tilde{g}$$ and $$\tilde{h}$$ are antiderivatives of $$g$$ and $$h$$, respectively.

b. Use the previous result to show that $$\displaystyle\iiint_B z \sin \sqrt{x^2 + y^2} \,dx \space dy \space dz = - 12 \pi^2,$$ where $$B$$ is a cylindrical shell with inner radius $$\pi$$ outer radius $$2\pi$$, and height $$2$$.

In exercises 9 - 12, the boundaries of the solid $$E$$ are given in cylindrical coordinates.

a. Express the region $$E$$ in cylindrical coordinates.

b. Convert the integral $$\displaystyle \iiint_E f(x,y,z) \,dV$$ to cylindrical coordinates.

9. $$E$$ is bounded by the right circular cylinder $$r = 4 \sin \theta$$, the $$r\theta$$-plane, and the sphere $$r^2 + z^2 = 16$$.

a. $$E = \big\{(r,\theta,z)\, | \,0 \leq \theta \leq \pi, \space 0 \leq r \leq 4 \sin \theta, \space 0 \leq z \leq \sqrt{16 - r^2}\big\}$$

b. $$\displaystyle\int_0^{\pi} \int_0^{4 \sin \theta} \int_0^{\sqrt{16-r^2}} f(r,\theta, z) r \, dz \space dr \space d\theta$$

10. $$E$$ is bounded by the right circular cylinder $$r = \cos \theta$$, the $$r\theta$$-plane, and the sphere $$r^2 + z^2 = 9$$.

11. $$E$$ is located in the first octant and is bounded by the circular paraboloid $$z = 9 - 3r^2$$, the cylinder $$r = \sqrt{3}$$, and the plane $$r(\cos \theta + \sin \theta) = 20 - z$$.

a. $$E = \big\{(r,\theta,z) \, | \, 0 \leq \theta \leq \frac{\pi}{2}, \space 0 \leq r \leq \sqrt{3}, \space 9 - r^2 \leq z \leq 10 - r(\cos \theta + \sin \theta)\big\}$$

b. $$\displaystyle\int_0^{\pi/2} \int_0^{\sqrt{3}} \int_{9-r^2}^{10-r(\cos \theta + \sin \theta)} f(r,\theta,z) r \space dz \space dr \space d\theta$$

12. $$E$$ is located in the first octant outside the circular paraboloid $$z = 10 - 2r^2$$ and inside the cylinder $$r = \sqrt{5}$$ and is bounded also by the planes $$z = 20$$ and $$\theta = \frac{\pi}{4}$$.

In exercises 13 - 16, the function $$f$$ and region $$E$$ are given.

a. Express the region $$E$$ and the function $$f$$ in cylindrical coordinates.

b. Convert the integral $$\displaystyle \iiint_B f(x,y,z) \,dV$$ into cylindrical coordinates and evaluate it.

13. $$f(x,y,z) = x^2 + y^2$$, $$E = \big\{(x,y,z)\, | \,0 \leq x^2 + y^2 \leq 9, \space x \geq 0, \space y \geq 0, \space 0 \leq z \leq x + 3\big\}$$

a. $$E = \big\{(r,\theta,z)\, | \,0 \leq r \leq 3, \space 0 \leq \theta \leq \frac{\pi}{2}, \space 0 \leq z \leq r \space \cos \theta + 3\big\},$$
$$f(r,\theta,z) = \frac{1}{r \space \cos \theta + 3}$$

b. $$\displaystyle \int_0^3 \int_0^{\pi/2} \int_0^{r \space \cos \theta+3} \frac{r}{r \space \cos \theta + 3} \, dz \space d\theta \space dr = \frac{9\pi}{4}$$

14. $$f(x,y,z) = x^2 + y^2, \space E = \big\{(x,y,z) |0 \leq x^2 + y^2 \leq 4, \space y \geq 0, \space 0 \leq z \leq 3 - x \big\}$$

15. $$f(x,y,z) = x, \space E = \big\{(x,y,z)\, | \,1 \leq y^2 + z^2 \leq 9, \space 0 \leq x \leq 1 - y^2 - z^2\big\}$$

a. $$y = r \space \cos \theta, \space z = r \space \sin \theta, \space x = z,\space E = \big\{(r,\theta,z)\, | \,1 \leq r \leq 3, \space 0 \leq \theta \leq 2\pi, \space 0 \leq z \leq 1 - r^2\big\}, \space f(r,\theta,z) = z$$;

b. $$\displaystyle \int_1^3 \int_0^{2\pi} \int_0^{1-r^2} z r \space dz \space d\theta \space dr = \frac{356 \pi}{3}$$

16. $$f(x,y,z) = y, \space E = \big\{(x,y,z)\, | \,1 \leq x^2 + z^2 \leq 9, \space 0 \leq y \leq 1 - x^2 - z^2 \big\}$$

In exercises 17 - 24, find the volume of the solid $$E$$ whose boundaries are given in rectangular coordinates.

17. $$E$$ is above the $$xy$$-plane, inside the cylinder $$x^2 + y^2 = 1$$, and below the plane $$z = 1$$.

$$\pi$$

18. $$E$$ is below the plane $$z = 1$$ and inside the paraboloid $$z = x^2 + y^2$$.

19. $$E$$ is bounded by the circular cone $$z = \sqrt{x^2 + y^2}$$ and $$z = 1$$.

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

20. $$E$$ is located above the $$xy$$-plane, below $$z = 1$$, outside the one-sheeted hyperboloid $$x^2 + y^2 - z^2 = 1$$, and inside the cylinder $$x^2 + y^2 = 2$$.

21. $$E$$ is located inside the cylinder $$x^2 + y^2 = 1$$ and between the circular paraboloids $$z = 1 - x^2 - y^2$$ and $$z = x^2 + y^2$$.

$$\pi$$

22. $$E$$ is located inside the sphere $$x^2 + y^2 + z^2 = 1$$, above the $$xy$$-plane, and inside the circular cone $$z = \sqrt{x^2 + y^2}$$.

23. $$E$$ is located outside the circular cone $$x^2 + y^2 = (z - 1)^2$$ and between the planes $$z = 0$$ and $$z = 2$$.

$$\frac{4\pi}{3}$$

24. $$E$$ is located outside the circular cone $$z = 1 - \sqrt{x^2 + y^2}$$, above the $$xy$$-plane, below the circular paraboloid, and between the planes $$z = 0$$ and $$z = 2$$.

25. [T] Use a computer algebra system (CAS) to graph the solid whose volume is given by the iterated integral in cylindrical coordinates $$\displaystyle \int_{-\pi/2}^{\pi/2} \int_0^1 \int_{r^2}^r r \, dz \, dr \, d\theta.$$  Find the volume $$V$$ of the solid. Round your answer to four decimal places.

$$V = \frac{pi}{12} \approx 0.2618$$

26. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in cylindrical coordinates $$\displaystyle \int_0^{\pi/2} \int_0^1 \int_{r^4}^r r \, dz \, dr \, d\theta.$$  Find the volume $$E$$ of the solid. Round your answer to four decimal places.

27. Convert the integral $$\displaystyle\int_0^1 \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} \int_{x^2+y^2}^{\sqrt{x^2+y^2}} xz \space dz \space dx \space dy$$ into an integral in cylindrical coordinates.

$$\displaystyle\int_0^1 \int_0^{\pi} \int_{r^2}^r zr^2 \space \cos \theta \, dz \space d\theta \space dr$$

28. Convert the integral $$\displaystyle \int_0^2 \int_0^y \int_0^1 (xy + z) \, dz \space dx \space dy$$ into an integral in cylindrical coordinates.

In exercises 29 - 32, evaluate the triple integral $$\displaystyle \iiint_B f(x,y,z) \,dV$$ over the solid $$B$$.

29. $$f(x,y,z) = 1, \space B = \big\{(x,y,z)\, | \,x^2 + y^2 + z^2 \leq 90, \space z \geq 0\big\}$$

[Hide Solution]

$$180 \pi \sqrt{10}$$

30. $$f(x,y,z) = 1 - \sqrt{x^2 + y^2 + z^2}, \space B = \big\{(x,y,z)\, | \,x^2 + y^2 + z^2 \leq 9, \space y \geq 0, \space z \geq 0\big\}$$

31. $$f(x,y,z) = \sqrt{x^2 + y^2}, \space B$$ is bounded above by the half-sphere $$x^2 + y^2 + z^2 = 9$$ with $$z \geq 0$$ and below by the cone $$2z^2 = x^2 + y^2$$.

$$\frac{81\pi(\pi - 2)}{16}$$

32. $$f(x,y,z) = \sqrt{x^2 + y^2}, \space B$$ is bounded above by the half-sphere $$x^2 + y^2 + z^2 = 16$$ with $$z \geq 0$$ and below by the cone $$2z^2 = x^2 + y^2$$.

33. Show that if $$F ( \rho,\theta,\varphi) = f(\rho)g(\theta)h(\varphi)$$ is a continuous function on the spherical box $$B = \big\{(\rho,\theta,\varphi)\, | \,a \leq \rho \leq b, \space \alpha \leq \theta \leq \beta, \space \gamma \leq \varphi \leq \psi\big\}$$, then $$\displaystyle\iiint_B F \space dV = \left(\int_a^b \rho^2 f(\rho) \space dr \right) \left( \int_{\alpha}^{\beta} g (\theta) \space d\theta \right)\left( \int_{\gamma}^{\psi} h (\varphi) \space \sin \varphi \space d\varphi \right).$$

34. A function $$F$$ is said to have spherical symmetry if it depends on the distance to the origin only, that is, it can be expressed in spherical coordinates as $$F(x,y,z) = f(\rho)$$, where $$\rho = \sqrt{x^2 + y^2 + z^2}$$. Show that $$\displaystyle\iiint_B F(x,y,z) \,dV = 2\pi \int_a^b \rho^2 f(\rho) \,d\rho,$$ where $$B$$ is the region between the upper concentric hemispheres of radii $$a$$ and $$b$$ centered at the origin, with $$0 < a < b$$ and $$F$$ a spherical function defined on $$B$$.

Use the previous result to show that $$\displaystyle\iiint_B (x^2 + y^2 + z^2) \sqrt{x^2 + y^2 + z^2} dV = 21 \pi,$$ where $$B = \big\{(x,y,z)\, | \,1 \leq x^2 + y^2 + z^2 \leq 2, \space z \geq 0\big\}$$.

35. Let $$B$$ be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where $$0 < a < b$$. Consider F a function defined on B whose form in spherical coordinates $$(\rho,\theta,\varphi)$$ is $$F(x,y,z) = f(\rho)\cos \varphi$$. Show that if $$g(a) = g(b) = 0$$ and $$\displaystyle\int_a^b h (\rho) \, d\rho = 0,$$ then $$\displaystyle\iiint_B F(x,y,z)\,dV = \frac{\pi^2}{4} [ah(a) - bh(b)],$$ where $$g$$ is an antiderivative of $$f$$ and $$h$$ is an antiderivative of $$g$$.

Use the previous result to show that $$\displaystyle \iiint_B = \frac{z \cos \sqrt{x^2 + y^2 + z^2}}{\sqrt{x^2 + y^2 + z^2}} \, dV = \frac{3\pi^2}{2},$$ where $$B$$ is the region between the upper concentric hemispheres of radii $$\pi$$ and $$2\pi$$ centered at the origin and situated in the first octant.

In exercises 36 - 39, the function $$f$$ and region $$E$$ are given.

a. Express the region $$E$$ and function $$f$$ in cylindrical coordinates.

b. Convert the integral $$\displaystyle \iiint_B f(x,y,z)\, dV$$ into cylindrical coordinates and evaluate it.

36. $$f(x,y,z) = z; \space E = \big\{(x,y,z)\, | \,0 \leq x^2 + y^2 + z^2 \leq 1, \space z \geq 0\big\}$$

37. $$f(x,y,z) = x + y; \space E = \big\{(x,y,z)\, | \,1 \leq x^2 + y^2 + z^2 \leq 2, \space z \geq 0, \space y \geq 0\big\}$$

a. $$f(\rho,\theta, \varphi) = \rho \space \sin \varphi \space (\cos \theta + \sin \theta), \space E = \big\{(\rho,\theta,\varphi)\, | \,1 \leq \rho \leq 2, \space 0 \leq \theta \leq \pi, \space 0 \leq \varphi \leq \frac{\pi}{2}\big\}$$;

b. $$\displaystyle \int_0^{\pi} \int_0^{\pi/2} \int_1^2 \rho^3 \cos \varphi \space \sin \varphi \space d\rho \space d\varphi \space d\theta = \frac{15\pi}{8}$$

38. $$f(x,y,z) = 2xy; \space E = \big\{(x,y,z)\, | \,\sqrt{x^2 + y^2} \leq z \leq \sqrt{1 - x^2 - y^2}, \space x \geq 0, \space y \geq 0\big\}$$

39. $$f(x,y,z) = z; \space E = \big\{(x,y,z)\, | \,x^2 + y^2 + z^2 - 2x \leq 0, \space \sqrt{x^2 + y^2} \leq z\big\}$$

a. $$f(\rho,\theta,\varphi) = \rho \space \cos \varphi; \space E = \big\{(\rho,\theta,\varphi)\, | \,0 \leq \rho \leq 2 \space \cos \varphi, \space 0 \leq \theta \leq \frac{\pi}{2}, \space 0 \leq \varphi \leq \frac{\pi}{4}\big\}$$;

b. $$\displaystyle\int_0^{\pi/2} \int_0^{\pi/4} \int_0^{2 \space \cos \varphi} \rho^3 \sin \varphi \space \cos \varphi \space d\rho \space d\varphi \space d\theta = \frac{7\pi}{24}$$

In exercises 40 - 41, find the volume of the solid $$E$$ whose boundaries are given in rectangular coordinates.

40. $$E = \big\{ (x,y,z)\, | \,\sqrt{x^2 + y^2} \leq z \leq \sqrt{16 - x^2 - y^2}, \space x \geq 0, \space y \geq 0\big\}$$

41. $$E = \big\{ (x,y,z)\, | \,x^2 + y^2 + z^2 - 2z \leq 0, \space \sqrt{x^2 + y^2} \leq z\big\}$$

$$\frac{\pi}{4}$$

42. Use spherical coordinates to find the volume of the solid situated outside the sphere $$\rho = 1$$ and inside the sphere $$\rho = \cos \varphi$$, with $$\varphi \in [0,\frac{\pi}{2}]$$.

43. Use spherical coordinates to find the volume of the ball $$\rho \leq 3$$ that is situated between the cones $$\varphi = \frac{\pi}{4}$$ and $$\varphi = \frac{\pi}{3}$$.

$$9\pi (\sqrt{2} - 1)$$

44. Convert the integral $$\displaystyle \int_{-4}^4 \int_{-\sqrt{16-y^2}}^{\sqrt{16-y^2}} \int_{-\sqrt{16-x^2-y^2}}^{\sqrt{16-x^2-y^2}} (x^2 + y^2 + z^2) \, dz \, dx \, dy$$ into an integral in spherical coordinates.

45. Convert the integral $$\displaystyle \int_0^4 \int_0^{\sqrt{16-x^2}} \int_{-\sqrt{16-x^2-y^2}}^{\sqrt{16-x^2-y^2}} (x^2 + y^2 + z^2)^2 \, dz \space dy \space dx$$ into an integral in spherical coordinates.

$$\displaystyle\int_0^{\pi/2} \int_0^{\pi/2} \int_0^4 \rho^6 \sin \varphi \, d\rho \, d\phi \, d\theta$$

47. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates $$\displaystyle \int_{\pi/2}^{\pi} \int_{5\pi}^{\pi/6} \int_0^2 \rho^2 \sin \varphi \space d\rho \space d\varphi \space d\theta.$$ Find the volume $$V$$ of the solid. Round your answer to three decimal places.

$$V = \frac{4\pi\sqrt{3}}{3} \approx 7.255$$

48. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates as $$\displaystyle \int_0^{2\pi} \int_{3\pi/4}^{\pi/4} \int_0^1 \rho^2 \sin \varphi \space d\rho \space d\varphi \space d\theta.$$ Find the volume $$V$$ of the solid. Round your answer to three decimal places.

49. [T] Use a CAS to evaluate the integral $$\displaystyle \iiint_E (x^2 + y^2) \, dV$$ where $$E$$ lies above the paraboloid $$z = x^2 + y^2$$ and below the plane $$z = 3y$$.

$$\frac{343\pi}{32}$$

50. [T]

a. Evaluate the integral $$\displaystyle \iiint_E e^{\sqrt{x^2+y^2+z^2}}\, dV,$$ where $$E$$ is bounded by spheres $$4x^2 + 4y^2 + 4z^2 = 1$$ and $$x^2 + y^2 + z^2 = 1$$.

b. Use a CAS to find an approximation of the previous integral. Round your answer to two decimal places.

51. Express the volume of the solid inside the sphere $$x^2 + y^2 + z^2 = 16$$ and outside the cylinder $$x^2 + y^2 = 4$$ as triple integrals in cylindrical coordinates and spherical coordinates, respectively.

$$\displaystyle \int_0^{2\pi}\int_2^4\int_{−\sqrt{16−r^2}}^{\sqrt{16−r^2}}r\,dz\,dr\,dθ$$  and  $$\displaystyle \int_{\pi/6}^{5\pi/6}\int_0^{2\pi}\int_{2\csc \phi}^{4}\rho^2\sin \rho \, d\rho \, d\theta \, d\phi$$

52. Express the volume of the solid inside the sphere $$x^2 + y^2 + z^2 = 16$$ and outside the cylinder $$x^2 + y^2 = 4$$ that is located in the first octant as triple integrals in cylindrical coordinates and spherical coordinates, respectively.

53. The power emitted by an antenna has a power density per unit volume given in spherical coordinates by $$p(\rho,\theta,\varphi) = \frac{P_0}{\rho^2} \cos^2 \theta \space \sin^4 \varphi$$, where $$P_0$$ is a constant with units in watts. The total power within a sphere $$B$$ of radius $$r$$ meters is defined as $$\displaystyle P = \iiint_B p(\rho,\theta,\varphi) \, dV.$$ Find the total power $$P$$.

$$P = \frac{32P_0 \pi}{3}$$ watts

54. Use the preceding exercise to find the total power within a sphere $$B$$ of radius 5 meters when the power density per unit volume is given by $$p(\rho, \theta,\varphi) = \frac{30}{\rho^2} \cos^2 \theta \sin^4 \varphi$$.

55. A charge cloud contained in a sphere $$B$$ of radius $$r$$ centimeters centered at the origin has its charge density given by $$q(x,y,z) = k\sqrt{x^2 + y^2 + z^2}\frac{\mu C}{cm^3}$$, where $$k > 0$$.  The total charge contained in $$B$$ is given by $$\displaystyle Q = \iiint_B q(x,y,z) \, dV.$$ Find the total charge $$Q$$.

$$Q = kr^4 \pi \mu C$$

56. Use the preceding exercise to find the total charge cloud contained in the unit sphere if the charge density is $$q(x,y,z) = 20 \sqrt{x^2 + y^2 + z^2} \frac{\mu C}{cm^3}$$.