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# 15.4E: Ejercicios para la Sección 15.4

• Edwin “Jed” Herman & Gilbert Strang
• OpenStax

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En los ejercicios 1 - 4, evalúe las integrales triples sobre la caja sólida rectangular$$B$$.

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

Responder
$$192$$

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

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

Responder
$$0$$

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

En los ejercicios 5 a 8, cambiar el orden de integración integrando primero con respecto a$$z$$, luego$$x$$, después$$y$$.

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

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

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

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

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

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

9. Dejar$$F$$,$$G$$, y$$H$$ ser funciones continuas sobre$$[a,b]$$,$$[c,d]$$, y$$[e,f]$$, respectivamente, dónde$$a, \, b, \, c, \, d, \, e$$, y$$f$$ son números reales tales que$$a < b, \, c < d$$, y$$e < f$$. Demostrar que

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

10. Dejar$$F$$,$$G$$, y$$H$$ ser funciones diferenciales sobre$$[a,b]$$$$[c,d]$$,, y$$[e,f]$$, respectivamente, dónde$$a, \, b, \, c, \, d, \, e$$, y$$f$$ son números reales tales que$$a < b, \, c < d$$, y$$e < f$$. Demostrar que

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

En los ejercicios 11 - 14, evaluar las triples integrales sobre la región delimitada

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

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

Responder
$$\frac{77}{12}$$

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

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

Responder
$$2$$

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

En los ejercicios 15 - 18, evaluar las triples integrales sobre la región delimitada indicada$$E$$.

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

Responder
$$\frac{430}{120}$$

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

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

Responder
$$0$$

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

En los ejercicios 19 - 22, evaluar las integrales triples sobre la región delimitada$$E$$ de la forma

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

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

Responder
$$-\frac{64}{105}$$

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

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

Responder
$$\frac{11}{26}$$

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

En los ejercicios 23 - 26, evaluar las triples integrales sobre la región delimitada

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

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

Responder
$$\frac{113}{450}$$

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

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

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

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

En los ejercicios 27 - 30, evaluar las triples integrales sobre la región delimitada

$$E = \big\{(x,y,z) \,|\, (x,y) \in D, \, u_1 (x,y) x \leq z \leq u_2 (x,y) \big\}$$, donde$$D$$ está la proyección de$$E$$ sobre el$$xy$$ plano.

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

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

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

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

Responder
$$1250$$

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

31. El sólido$$E$$ delimitado por$$y^2 + z^2 = 9, \, z = 0$$, y$$x = 5$$ se muestra en la siguiente figura. Evaluar la integral$$\displaystyle \iiint_E z \, dV$$ integrando primero con respecto a$$z$$, luego$$y$$, y luego$$x$$.

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

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

33. [T] El volumen de un sólido$$E$$ viene dado por la integral$$\displaystyle \int_{-2}^0 \int_x^0 \int_0^{x^2+y^2} \, dz \, dy \, dx.$$ Utilice un sistema de álgebra computacional (CAS) para graficar$$E$$ y encontrar su volumen. Redondea tu respuesta a dos decimales.

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

34. [T] El volumen de un sólido$$E$$ 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.

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

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

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 $$a > 0$$.

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

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 $$z = 5$$.

$$\frac{9}{2}$$

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 $$x + y = 7$$.

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

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 $$x = \frac{\pi}{2}$$.

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.

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 following figure.

a.- Escribir la triple integral que da el volumen de$$E$$ integrando primero con respecto a$$z$$, luego con$$y$$, y luego con$$x$$.

b. reescribir la integral en la parte a. como integral equivalente en otros cinco órdenes.

Responder

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. El sólido$$E$$ delimitado por la esfera de ecuación$$9x^2 + 4y^2 + z^2 = 1$$ y ubicado en el primer octante se representa en la siguiente figura.

a. escribir la triple integral que da el volumen de$$E$$ 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 $$(2,0,1)$$.

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

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 following figure. Find the volume of the solid.

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

52. El sólido$$E$$ delimitado por$$z = 1 - x^2$$ y situado en el primer octante se da en la siguiente figura. Encuentra el volumen del sólido.

### Aproximación a triples integrales

53. La regla del punto medio para la 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.

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

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.

$$17.5^{\circ}$$ 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 $$E$$.

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