2.6E: Exercises

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

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

Answer

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

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

Answer

$\frac{1}{8}$

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

Answer

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

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 $r^2 + z^2 = 16$.

Answer

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 $r(\cos \theta + \sin \theta) = 20 - z$.

Answer

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

Answer

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

Answer

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 $z = 1$.

Answer

$\pi$

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 $z = 1$.

Answer

$\frac{\pi}{3}$

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 $z = x^2 + y^2$.

Answer

$\pi$

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 $z = 2$.

Answer

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

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.

Answer

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

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.

Answer

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

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]

Answer

$180 \pi \sqrt{10}$

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 $2z^2 = x^2 + y^2$.

Answer

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

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

Answer

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

Answer

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

Answer

$\frac{\pi}{4}$

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 $\varphi = \frac{\pi}{3}$.

Answer

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

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.

Answer

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

46. Convert the integral $\displaystyle \int_{-2}^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int_{\sqrt{x^2+y^2}}^{\sqrt{16-x^2-y^2}} \, dz \space dy \space dx$ into an integral in spherical coordinates and evaluate it.

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.

Answer

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

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 $z = 3y$.

Answer

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

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.

Answer

$\displaystyle \int_0^{2\pi}\int_2^4\int_{−\sqrt{16−r^2}}^{\sqrt{16−r^2}}r\,dz\,dr\,dθ$ 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 $P$.

Answer

$P = \frac{32P_0 \pi}{3}$ 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 $Q$.

Answer

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

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

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.