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# 14.6E: Exercises for Section 14.6

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In exercises 1 - 12, the region $$R$$ occupied by a lamina is shown in a graph. Find the mass of $$R$$ with the density function $$\rho$$.

1. $$R$$ is the triangular region with vertices $$(0,0), \space (0,3)$$, and $$(6,0); \space \rho (x,y) = xy$$. $$\frac{27}{2}$$

2. $$R$$ is the triangular region with vertices $$(0,0), \space (1,1)$$, and $$(0,5); \space \rho (x,y) = x + y$$. 3. $$R$$ is the rectangular region with vertices $$(0,0), \space (0,3), \space (6,3)$$ and $$(6,0); \space \rho (x,y) = \sqrt{xy}$$. $$24\sqrt{2}$$

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. $$R$$ is the trapezoidal region determined by the lines $$y = - \frac{1}{4}x + \frac{5}{2}, \space y = 0, \space y = 2$$, and $$x = 0; \space \rho (x,y) = 3xy$$. $$76$$

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

$$8\pi$$

8. $$R$$ is the unit disk; $$\rho (x,y) = 3x^4 + 6x^2y^2 + 3y^4$$. 9. $$R$$ is the region enclosed by the ellipse $$x^2 + 4y^2 = 1; \space \rho(x,y) = 1$$. $$\frac{\pi}{2}$$

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 $$y = x, \space y = -x, \space y = x + 2, \space y = -x + 2; \space \rho(x,y) = 1$$. $$2$$

12. $$R$$ 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 13 - 24, consider a lamina occupying the region $$R$$ and having the density function $$\rho$$ given in the preceding group of exercises. Use a computer algebra system (CAS) to answer the following questions.

a. Find the moments $$M_x$$ and $$M_y$$ about the $$x$$-axis and $$y$$-axis, respectively.

b. Calculate and plot the center of mass of the lamina.

c. [T] Use a CAS to locate the center of mass on the graph of $$R$$.

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

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

a. $$M_x = \frac{216\sqrt{2}}{5}, \space M_y = \frac{432\sqrt{2}}{5}$$;
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] $$R$$ is the trapezoidal region determined by the lines $$y = 0, \space y = 1, \space y = x,$$ and $$y = -x + 3; \space \rho (x,y) = 2x + y$$.

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

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 $$y = -x + 2; \space \rho (x,y) = 1$$.

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. Consider the solid $$Q = \big\{ (x,y,z) \,|\, 0 \leq x \leq 1, \space 0 \leq y \leq 2, \space 0 \leq z \leq 3\big\}$$ with the density function $$\rho(x,y,z) = x + y + 1$$.

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 $$Q$$.

43. [T] The solid $$Q$$ has the mass given by the triple integral $$\displaystyle \int_{-1}^1 \int_0^{\pi/4} \int_0^1 r^2 \, dr \space d\theta \space dz.$$

Use a CAS to answer the following questions.

• Show that the center of mass of $$Q$$ is located in the $$xy$$-plane.
• Graph $$Q$$ and locate its center of mass.

$$\bar{x} = \frac{3\sqrt{2}}{2\pi}$$, $$\bar{y} = \frac{3(2-\sqrt{2})}{2\pi}, \space \bar{z} = 0$$; 2. the solid $$Q$$ and its center of mass are shown in the following figure. 44. The solid $$Q$$ is bounded by the planes $$x + 4y + z = 8, \space x = 0, \space y = 0$$, and $$z = 0$$. Its density at any point is equal to the distance to the $$xz$$-plane. Find the moments of inertia of the solid about the $$xz$$-plane.

45. The solid $$Q$$ is bounded by the planes $$x + y + z = 3, \space x = 0, \space y = 0$$, and $$z = 0$$. Its density is $$\rho(x,y,z) = x + ay$$, where $$a > 0$$. Show that the center of mass of the solid is located in the plane $$z = \frac{3}{5}$$ for any value of $$a$$.

46. Let $$Q$$ be the solid situated outside the sphere $$x^2 + y^2 + z^2 = z$$ and inside the upper hemisphere $$x^2 + y^2 + z^2 = R^2$$, where $$R > 1$$. If the density of the solid is $$\rho (x,y,z) = \frac{1}{\sqrt{x^2+y^2+z^2}}$$, find $$R$$ such that the mass of the solid is $$\frac{7\pi}{2}.$$

47. The mass of a solid $$Q$$ is given by $$\displaystyle \int_0^2 \int_0^{\sqrt{4-x^2}} \int_{\sqrt{x^2+y^2}}^{\sqrt{16-x^2-y^2}} (x^2 + y^2 + z^2)^n \, dz \space dy \space dx,$$ where $$n$$ is an integer. Determine $$n$$ such the mass of the solid is $$(2 - \sqrt{2}) \pi$$.

$$n = -2$$

48. Let $$Q$$ be the solid bounded above the cone $$x^2 + y^2 = z^2$$ and below the sphere $$x^2 + y^2 + z^2 - 4z = 0$$. Its density is a constant $$k > 0$$. Find $$k$$ such that the center of mass of the solid is situated $$7$$ units from the origin.

49. The solid $$Q = \big\{(x,y,z) \,|\, 0 \leq x^2 + y^2 \leq 16, \space x \geq 0, \space y \geq 0, \space 0 \leq z \leq x\big\}$$ has the density $$\rho (x,y,z) = k$$. Show that the moment $$M_{xy}$$ about the $$xy$$-plane is half of the moment $$M_{yz}$$ about the $$yz$$-plane.

50. The solid $$Q$$ is bounded by the cylinder $$x^2 + y^2 = a^2$$, the paraboloid $$b^2 - z = x^2 + y^2$$, and the $$xy$$-plane, where $$0 < a < b$$. Find the mass of the solid if its density is given by $$\rho(x,y,z) = \sqrt{x^2 + y^2}$$.

51. Let $$Q$$ be a solid of constant density $$k$$, where $$k > 0$$, that is located in the first octant, inside the circular cone $$x^2 + y^2 = 9(z - 1)^2$$, and above the plane $$z = 0$$. Show that the moment $$M_{xy}$$ about the $$xy$$-plane is the same as the moment $$M_{yz}$$ about the $$xz$$-plane.

52. The solid $$Q$$ has the mass given by the triple integral $$\displaystyle \int_0^1 \int_0^{\pi/2} \int_0^{r^3} (r^4 + r) \space dz \space d\theta \space dr.$$

a. Find the density of the solid in rectangular coordinates.

b. Find the moment $$M_{xy}$$ about the $$xy$$-plane.

53. The solid $$Q$$ has the moment of inertia $$I_x$$ about the $$yz$$-plane given by the triple integral $$\displaystyle \int_0^2 \int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}} \int_{\frac{1}{2}(x^2+y^2)}^{\sqrt{x^2+y^2}} (y^2 + z^2)(x^2 + y^2) \, dz \space dx \space dy.$$

a. Find the density of $$Q$$.

b. Find the moment of inertia $$I_z$$ about the $$xy$$-plane.

a. $$\rho (x,y,z) = x^2 + y^2$$;
b. $$\frac{16\pi}{7}$$

54. The solid $$Q$$ has the mass given by the triple integral $$\displaystyle \int_0^{\pi/4} \int_0^{2 \space sec \space \theta} \int_0^1 (r^3 \cos \theta \sin \theta + 2r) \, dz \space dr \space d\theta.$$

a. Find the density of the solid in rectangular coordinates.

b. Find the moment $$M_{xz}$$ about the $$xz$$-plane.

55. Let $$Q$$ be the solid bounded by the $$xy$$-plane, the cylinder $$x^2 + y^2 = a^2$$, and the plane $$z = 1$$, where $$a > 1$$ is a real number. Find the moment $$M_{xy}$$ of the solid about the $$xy$$-plane if its density given in cylindrical coordinates is $$\rho(x,y,z) = \frac{d^2f}{dr^2} (r)$$, where $$f$$ is a differentiable function with the first and second derivatives continuous and differentiable on $$(0,a)$$.

$$M_{xy} = \pi (f(0) - f(a) + af'(a))$$

56. A solid $$Q$$ has a volume given by $$\displaystyle \iint_D \int_a^b \, dA \space dz$$, where $$D$$ is the projection of the solid onto the $$xy$$-plane and $$a < b$$ are real numbers, and its density does not depend on the variable $$z$$. Show that its center of mass lies in the plane $$z = \frac{a+b}{2}$$.

57. Consider the solid enclosed by the cylinder $$x^2 + z^2 = a^2$$ and the planes $$y = b$$ and $$y = c$$, where $$a > 0$$ and $$b < c$$ are real numbers. The density of $$Q$$ is given by $$\rho(x,y,z) = f'(y)$$, where $$f$$ is a differential function whose derivative is continuous on $$(b,c)$$. Show that if $$f(b) = f(c)$$, then the moment of inertia about the $$xz$$-plane of $$Q$$ is null.

58. [T] The average density of a solid $$Q$$ is defined as $$\displaystyle \rho_{ave} = \frac{1}{V(Q)} \iiint_Q \rho(x,y,z) \, dV = \frac{m}{V(Q)},$$ where $$V(Q)$$ and $$m$$ are the volume and the mass of $$Q$$, respectively. If the density of the unit ball centered at the origin is $$\rho (x,y,z) = e^{-x^2-y^2-z^2}$$, use a CAS to find its average density. Round your answer to three decimal places.

59. Show that the moments of inertia $$I_x, \space I_y$$, and $$I_z$$ about the $$yz$$-plane, $$xz$$-plane, and $$xy$$-plane, respectively, of the unit ball centered at the origin whose density is $$\rho (x,y,z) = e^{-x^2-y^2-z^2}$$ are the same. Round your answer to two decimal places.

$$I_x = I_y = I_z \approx 0.84$$