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# 7.6E

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### Exercise $$\PageIndex{1}$$: Mass

In the following exercises, 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 $$(6,0); \space \rho (x,y) = xy$$.

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

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

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

$$24\sqrt{2}$$

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

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

$$76$$

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

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

$$8\pi$$

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

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

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

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

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

$$2$$

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

### Exercise $$\PageIndex{2}$$ (CAS)

In the following exercises, 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 $$R$$.

1. [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.

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

3. [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.

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

[5. 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.

6. [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\).

7. [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.

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

8. [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.

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

10. [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.

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

### Exercise $$\PageIndex{3}$$

In the following exercises, 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.

1. $$R$$ 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 $$R_0 = 3$$

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 $$(6,0); \space \rho (x,y) = \sqrt{xy}$$.

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 $$R_0 = \frac{3\sqrt{106}}{7}$$

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\).

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 $$R_0 = \frac{2\sqrt{1957}}{19}$$

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 $$(1,2); \space \rho(x,y) = x^2 + y^2 - 2x - 4y + 5$$.

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 $$R_0 = \frac{\sqrt{69}}{3}$$

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 $$x^2 + 4y^2 = 1; \space \rho(x,y) = 1$$.

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 $$R_0 = \frac{\sqrt{5}}{4}$$

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

11. $$R$$ is the region bounded by $$y = x, \space y = -x, \space y = x + 2$$, 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 $$R_0 = \frac{2\sqrt{3}}{3}$$

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

### Exercise $$\PageIndex{4}$$: Mass of a solid

1. 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.

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

2. 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.

3. 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.

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

5. [T] The solid $$Q = \{ (x,y,z) | x^2 + y^2 \leq 9, \space 0 \leq z \leq 1, \space x \geq 0, \space y \geq 0\}$$ 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.

6. Consider the solid $$Q = \{ (x,y,z) | 0 \leq x \leq 1, \space 0 \leq y \leq 2, \space 0 \leq z \leq 3\}$$ 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 $$Q$$.

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

8. 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.

9. 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.

10. 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 $$a$$.

11. 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 $$\frac{7\pi}{2}.$$

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

$$n = -2$$

13. 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.

14. The solid $$Q = \{(x,y,z) | 0 \leq x^2 + y^2 \leq 16, \space x \geq 0, \space y \geq 0, \space 0 \leq z \leq x\}$$ 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.

15. 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 $$\rho(x,y,z) = \sqrt{x^2 + y^2}$$.

16. 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.

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

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

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

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

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. $$\frac{16\pi}{7}$$

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

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

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

20. 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 $$(0,a)$$.

$$M_{xy} = \pi (f(0) - f(a) + af'(a))$$
21. A solid $$Q$$ has a volume given by $\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 $$z = \frac{a+b}{2}$$. 22. 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. 23. [T] The average density of a solid $$Q$$ is defined as \[\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.
23. 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.
$$I_x = I_y = I_z \approx 0.84$$