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

A right triangle bounded by the x and y axes and the line y = negative x/2 + 3.

Answer: $\frac{27}{2}$

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

A triangle bounded by the y axis, the line x = y, and the line y = negative 4x + 5.

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

A rectangle bounded by the x and y axes and the lines x = 6 and y = 3.

Answer: $24\sqrt{2}$

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$.

A rectangle bounded by the y axis, the lines y = 1 and 3, and the line x = 3.

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 $x = 0; \space \rho (x,y) = 3xy$.

A trapezoid bounded by the x and y axes, the line y = 2, and the line y = negative x/4 + 2.5.

Answer: $76$

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

A trapezoid bounded by the x axis, the line y = 1, the line y = x, and the line y = negative x + 3.

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

$(1, 2);$

Answer: $8\pi$

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

A circle with radius 1 and center the origin.

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

An ellipse with center the origin, major axis 2, and minor axis 0.5.

Answer: $\frac{\pi}{2}$

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

The quarter section of an ellipse in the first quadrant with center the origin, major axis 2, and minor axis roughly 0.64.

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

A square with side length square root of 2 rotated 45 degrees, with corners at the origin, (2, 0), (1, 1), and (negative 1, 1).

Answer: $2$

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

A complex region between 2 and 1 that sweeps down and to the right with boundaries y = 1/x and y = 2/x.

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

Answer

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

A triangular region R bounded by the x and y axes and the line y = negative x/2 + 3, with a point marked at (12/5, 6/5).

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

Answer

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

A rectangle R bounded by the x and y axes and the lines x = 6 and y = 3 with point marked (18/5, 9/5).

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\).

Answer

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

A trapezoid R bounded by the x and y axes, the line y = 2, and the line y = negative x/4 + 2.5 with the point marked (92/95, 388/95).

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

Answer

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

A circle with radius 2 centered at (1, 2), which is tangent to the x axis at (1, 0) and has pointed marked at the center (1, 2).

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

Answer

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

An ellipse R with center the origin, major axis 2, and minor axis 0.5, with point marked at the origin.

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

Answer

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

A square R with side length square root of 2 rotated 45 degrees, with corners at the origin, (2, 0), (1, 1), and (negative 1, 1). A point is marked at (0, 1).

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

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

Answer: 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\).

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

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

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

Answer: 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.

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

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.

Answer

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. $\bar{x} = \frac{9}{2\pi}, \space \bar{y} = \frac{9}{2\pi}, \space \bar{z} = \frac{2}{3}$;

A quarter cylinder in the first quadrant with height 1 and radius 3. A point is marked at (9/(2 pi), 9/(2 pi), 2/3).

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.

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 $(2 - \sqrt{2}) \pi$.

Answer: $n = -2$

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 \[\int_0^1 \int_0^{\pi/2} \int_0^{r^3} (r^4 + r) \space dz \space d\theta \space dr.\]

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.

Answer: 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 $(0,a)$.

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

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.

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

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Exercise \(\PageIndex{1}\): Mass

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

Exercise \(\PageIndex{3}\)

Exercise \(\PageIndex{4}\): Mass of a solid