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.