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# 15.6E: Ejercicios para la Sección 15.6

• Edwin “Jed” Herman & Gilbert Strang
• OpenStax

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En los ejercicios 1 - 12, la región$$R$$ ocupada por una lámina se muestra en una gráfica. Encuentra la masa de$$R$$ con la función de densidad$$\rho$$.

1. $$R$$es la región triangular con vértices$$(0,0), \space (0,3)$$, y$$(6,0); \space \rho (x,y) = xy$$.

Contestar
$$\frac{27}{2}$$

2. $$R$$ is the triangular region with vertices $$(0,0), \space (1,1)$$, and $$(0,5); \space \rho (x,y) = x + y$$.

3. $$R$$es la región rectangular con vértices$$(0,0), \space (0,3), \space (6,3)$$ y$$(6,0); \space \rho (x,y) = \sqrt{xy}$$.

Contestar
$$24\sqrt{2}$$

4. $$R$$ is the rectangular region with vertices $$(0,1), \space (0,3), \space (3,3)$$ and $$(3,1); \space \rho (x,y) = x^2y$$.

5. $$R$$es la región trapezoidal determinada por las líneas$$y = - \frac{1}{4}x + \frac{5}{2}, \space y = 0, \space y = 2$$, y$$x = 0; \space \rho (x,y) = 3xy$$.

Contestar
$$76$$

6. $$R$$ is the trapezoidal region determined by the lines $$y = 0, \space y = 1, \space y = x$$ and $$y = -x + 3; \space \rho (x,y) = 2x + y$$.

7. $$R$$es el disco de radio$$2$$ centrado en$$(1,2); \space \rho(x,y) = x^2 + y^2 - 2x - 4y + 5$$.

Contestar
$$8\pi$$

8. $$R$$es el disco de la unidad;$$\rho (x,y) = 3x^4 + 6x^2y^2 + 3y^4$$.

9. $$R$$ is the region enclosed by the ellipse $$x^2 + 4y^2 = 1; \space \rho(x,y) = 1$$.

Contestar
$$\frac{\pi}{2}$$

10. $$R = \big\{(x,y) \,|\, 9x^2 + y^2 \leq 1, \space x \geq 0, \space y \geq 0\big\} ; \space \rho (x,y) = \sqrt{9x^2 + y^2}$$.

11. $$R$$ is the region bounded by $$y = x, \space y = -x, \space y = x + 2, \space y = -x + 2; \space \rho(x,y) = 1$$.

Contestar
$$2$$

12. $$R$$es la región delimitada por$$y = \frac{1}{x}, \space y = \frac{2}{x}, \space y = 1$$, y$$y = 2; \space \rho (x,y) = 4(x + y)$$.

En los ejercicios 13 - 24, considere una lámina ocupando la región$$R$$ y teniendo la función de densidad$$\rho$$ dada en el grupo anterior de ejercicios. Utilice un sistema de álgebra computacional (CAS) para responder a las siguientes preguntas.

a. Encuentra los momentos$$M_x$$ y$$M_y$$ alrededor del$$x$$ eje -eje y$$y$$ -eje, respectivamente.

b. Calcular y trazar el centro de masa de la lámina.

c. [T] Utilice un CAS para ubicar el centro de masa en la gráfica de$$R$$.

13. [T]$$R$$ is the triangular region with vertices $$(0,0), \space (0,3)$$, and $$(6,0); \space \rho (x,y) = xy$$.

a. $$M_x = \frac{81}{5}, \space M_y = \frac{162}{5}$$;
b. $$\bar{x} = \frac{12}{5}, \space \bar{y} = \frac{6}{5}$$;
c.

14. [T]$$R$$ es la región triangular con vértices$$(0,0), \space (1,1)$$, y$$(0,5); \space \rho (x,y) = x + y$$.

15. [T]$$R$$ es la región rectangular con vértices$$(0,0), \space (0,3), \space (6,3)$$, y$$(6,0); \space \rho (x,y) = \sqrt{xy}$$.

Contestar

a.$$M_x = \frac{216\sqrt{2}}{5}, \space M_y = \frac{432\sqrt{2}}{5}$$;
b.$$\bar{x} = \frac{18}{5}, \space \bar{y} = \frac{9}{5}$$;
c.

16. [T]$$R$$ is the rectangular region with vertices $$(0,1), \space (0,3), \space (3,3)$$, and $$(3,1); \space \rho (x,y) = x^2y$$.

17. [T] $$R$$ is the trapezoidal region determined by the lines $$y = - \frac{1}{4}x + \frac{5}{2}, \space y = 0, \space y = 2$$, and $$x = 0; \space \rho (x,y) = 3xy$$.

a. $$M_x = \frac{368}{5}, \space M_y = \frac{1552}{5}$$;
b. $$\bar{x} = \frac{92}{95}, \space \bar{y} = \frac{388}{95}$$;
c.

18. [T]$$R$$ es la región trapezoidal determinada por las líneas$$y = 0, \space y = 1, \space y = x,$$ y$$y = -x + 3; \space \rho (x,y) = 2x + y$$.

19. [T]$$R$$ es el disco de radio$$2$$ centrado en$$(1,2); \space \rho(x,y) = x^2 + y^2 - 2x - 4y + 5$$.

Contestar

a.$$M_x = 16\pi, \space M_y = 8\pi$$;
b.$$\bar{x} = 1, \space \bar{y} = 2$$;
c.

20. [T]$$R$$ is the unit disk; $$\rho (x,y) = 3x^4 + 6x^2y^2 + 3y^4$$.

21. [T] $$R$$ is the region enclosed by the ellipse $$x^2 + 4y^2 = 1; \space \rho(x,y) = 1$$.

a. $$M_x = 0, \space M_y = 0)$$;
b. $$\bar{x} = 0, \space \bar{y} = 0$$;
c.

22. [T]$$R = \big\{(x,y) \,|\, 9x^2 + y^2 \leq 1, \space x \geq 0, \space y \geq 0\big\} ; \space \rho (x,y) = \sqrt{9x^2 + y^2}$$.

23. [T]$$R$$ is the region bounded by $$y = x, \space y = -x, \space y = x + 2$$, and $$y = -x + 2; \space \rho (x,y) = 1$$.

Contestar

a.$$M_x = 2, \space M_y = 0)$$;
b.$$\bar{x} = 0, \space \bar{y} = 1$$;
c.

24. [T]$$R$$ is the region bounded by $$y = \frac{1}{x}, \space y = \frac{2}{x}, \space y = 1$$, and $$y = 2; \space \rho (x,y) = 4(x + y)$$.

In exercises 25 - 36, consider a lamina occupying the region $$R$$ and having the density function $$\rho$$ given in the first two groups of Exercises.

a. Find the moments of inertia $$I_x, \space I_y$$ and $$I_0$$ about the $$x$$-axis, $$y$$-axis, and origin, respectively.

b. Find the radii of gyration with respect to the $$x$$-axis, $$y$$-axis, and origin, respectively.

25. $$R$$ is the triangular region with vertices $$(0,0), \space (0,3)$$, and $$(6,0); \space \rho (x,y) = xy$$.

a. $$I_x = \frac{243}{10}, \space I_y = \frac{486}{5}$$, and $$I_0 = \frac{243}{2}$$;
b. $$R_x = \frac{3\sqrt{5}}{5}, \space R_y = \frac{6\sqrt{5}}{5}$$, and $$R_0 = 3$$

26. $$R$$ is the triangular region with vertices $$(0,0), \space (1,1)$$, and $$(0,5); \space \rho (x,y) = x + y$$.

27. $$R$$ is the rectangular region with vertices $$(0,0), \space (0,3), \space (6,3)$$, and $$(6,0); \space \rho (x,y) = \sqrt{xy}$$.

a. $$I_x = \frac{2592\sqrt{2}}{7}, \space I_y = \frac{648\sqrt{2}}{7}$$, and $$I_0 = \frac{3240\sqrt{2}}{7}$$;
b. $$R_x = \frac{6\sqrt{21}}{7}, \space R_y = \frac{3\sqrt{21}}{7}$$, and $$R_0 = \frac{3\sqrt{106}}{7}$$

28. $$R$$ is the rectangular region with vertices $$(0,1), \space (0,3), \space (3,3)$$, and $$(3,1); \space \rho (x,y) = x^2y$$.

29. $$R$$ is the trapezoidal region determined by the lines $$y = - \frac{1}{4}x + \frac{5}{2}, \space y = 0, \space y = 2$$, and x = 0; \space \rho (x,y) = 3xy\).

a. $$I_x = 88, \space I_y = 1560$$, and $$I_0 = 1648$$;
b. $$R_x = \frac{\sqrt{418}}{19}, \space R_y = \frac{\sqrt{7410}}{10}$$, and $$R_0 = \frac{2\sqrt{1957}}{19}$$

30. $$R$$ is the trapezoidal region determined by the lines $$y = 0, \space y = 1, \space y = x$$, and y = -x + 3; \space \rho (x,y) = 2x + y\).

31. $$R$$ is the disk of radius $$2$$ centered at $$(1,2); \space \rho(x,y) = x^2 + y^2 - 2x - 4y + 5$$.

a. $$I_x = \frac{128\pi}{3}, \space I_y = \frac{56\pi}{3}$$, and $$I_0 = \frac{184\pi}{3}$$;
b. $$R_x = \frac{4\sqrt{3}}{3}, \space R_y = \frac{\sqrt{21}}{2}$$, and $$R_0 = \frac{\sqrt{69}}{3}$$

32. $$R$$ is the unit disk; $$\rho (x,y) = 3x^4 + 6x^2y^2 + 3y^4$$.

33. $$R$$ is the region enclosed by the ellipse $$x^2 + 4y^2 = 1; \space \rho(x,y) = 1$$.

a. $$I_x = \frac{\pi}{32}, \space I_y = \frac{\pi}{8}$$, and $$I_0 = \frac{5\pi}{32}$$;
b. $$R_x = \frac{1}{4}, \space R_y = \frac{1}{2}$$, and $$R_0 = \frac{\sqrt{5}}{4}$$

34. $$R = \big\{(x,y) \,|\, 9x^2 + y^2 \leq 1, \space x \geq 0, \space y \geq 0\big\} ; \space \rho (x,y) = \sqrt{9x^2 + y^2}$$.

35. $$R$$ is the region bounded by $$y = x, \space y = -x, \space y = x + 2$$, and $$y = -x + 2; \space \rho (x,y) = 1$$.

a. $$I_x = \frac{7}{3}, \space I_y = \frac{1}{3}$$, and $$I_0 = \frac{8}{3}$$;
b. $$R_x = \frac{\sqrt{42}}{6}, \space R_y = \frac{\sqrt{6}}{6}$$, and $$R_0 = \frac{2\sqrt{3}}{3}$$

36. $$R$$ is the region bounded by $$y = \frac{1}{x}, \space y = \frac{2}{x}, \space y = 1$$, and $$y = 2; \space \rho (x,y) = 4(x + y)$$.

37. Let $$Q$$ be the solid unit cube. Find the mass of the solid if its density $$\rho$$ is equal to the square of the distance of an arbitrary point of $$Q$$ to the $$xy$$-plane.

$$m = \frac{1}{3}$$

38. Let $$Q$$ be the solid unit hemisphere. Find the mass of the solid if its density $$\rho$$ is proportional to the distance of an arbitrary point of $$Q$$ to the origin.

39. The solid $$Q$$ of constant density $$1$$ is situated inside the sphere $$x^2 + y^2 + z^2 = 16$$ and outside the sphere $$x^2 + y^2 + z^2 = 1$$. Show that the center of mass of the solid is not located within the solid.

40. Find the mass of the solid $$Q = \big\{ (x,y,z) \,|\, 1 \leq x^2 + z^2 \leq 25, \space y \leq 1 - x^2 - z^2 \big\}$$ whose density is $$\rho (x,y,z) = k$$, where $$k > 0$$.

41. [T] The solid $$Q = \big\{ (x,y,z) \,|\, x^2 + y^2 \leq 9, \space 0 \leq z \leq 1, \space x \geq 0, \space y \geq 0\big\}$$ has density equal to the distance to the $$xy$$-plane. Use a CAS to answer the following questions.

a. Find the mass of $$Q$$.

b. Find the moments $$M_{xy}, \space M_{xz}$$ and $$M_{yz}$$ about the $$xy$$-plane, $$xz$$-plane, and $$yz$$-plane, respectively.

c. Find the center of mass of $$Q$$.

d. Graph $$Q$$ and locate its center of mass.

a. $$m = \frac{9\pi}{4}$$;
b. $$M_{xy} = \frac{3\pi}{2}, \space M_{xz} = \frac{81}{8}, \space M_{yz} = \frac{81}{8}$$;
c. $$\bar{x} = \frac{9}{2\pi}, \space \bar{y} = \frac{9}{2\pi}, \space \bar{z} = \frac{2}{3}$$;
d.

42. Considera el sólido$$Q = \big\{ (x,y,z) \,|\, 0 \leq x \leq 1, \space 0 \leq y \leq 2, \space 0 \leq z \leq 3\big\}$$ con la función de densidad$$\rho(x,y,z) = x + y + 1$$.

a. encontrar la masa de$$Q$$.

b. Encontrar los momentos$$M_{xy}, \space M_{xz}$$ y$$M_{yz}$$ sobre el$$xy$$ -plano,$$xz$$ -plano, y$$yz$$ -plano, respectivamente.

c. Encuentra el centro de masa de$$Q$$.

43. [T] El sólido$$Q$$ tiene la masa dada por la triple integral$$\displaystyle \int_{-1}^1 \int_0^{\pi/4} \int_0^1 r^2 \, dr \space d\theta \space dz.$$

Utilice un CAS para responder a las siguientes preguntas.

• Demostrar que el centro de masa de$$Q$$ está ubicado en el$$xy$$ -plano.
• Grafica$$Q$$ y localiza su centro de masa.
Contestar

$$\bar{x} = \frac{3\sqrt{2}}{2\pi}$$,$$\bar{y} = \frac{3(2-\sqrt{2})}{2\pi}, \space \bar{z} = 0$$; 2. el sólido$$Q$$ y su centro de masa se muestran en la siguiente figura.

44. El sólido

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