7.6E

Last updated

Nov 19, 2020
Save as PDF
- 7.6: Calculating Centers of Mass and Moments of Inertia
- 7.7: Change of Variables in Multiple Integrals

( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercise $1$ : 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 $ρ$ .

1. $R$ is the triangular region with vertices $(0, 0), (0, 3)$ , and $(6, 0); ρ (x, y) = x y$ .

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 $(0, 0), (1, 1)$ , and $(0, 5); ρ (x, y) = x + y$ .

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 $(0, 0), (0, 3), (6, 3)$ and $(6, 0); ρ (x, y) = \sqrt{x y}$ .

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 $(0, 1), (0, 3), (3, 3)$ and $(3, 1); ρ (x, y) = x^{2} y$ .

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 $y = - \frac{1}{4} x + \frac{5}{2}, y = 0, y = 2$ , and $x = 0; ρ (x, y) = 3 x y$ .

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 $y = 0, y = 1, y = x$ and $y = - x + 3; ρ (x, y) = 2 x + y$ .

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 $(1, 2); ρ (x, y) = x^{2} + y^{2} - 2 x - 4 y + 5$ .

$(1, 2);$

Answer: $8 π$

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

A circle with radius 1 and center the origin.

11. $R$ is the region enclosed by the ellipse $x^{2} + 4 y^{2} = 1; ρ (x, y) = 1$ .

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

Answer: $\frac{π}{2}$

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

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 $y = x, y = - x, y = x + 2, y = - x + 2; ρ (x, y) = 1$ .

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 $y = \frac{1}{x}, y = \frac{2}{x}, y = 1$ , and $y = 2; ρ (x, y) = 4 (x + y)$ .

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

Exercise $2$ (CAS)

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ 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 $(0, 0), (0, 3)$ , and $(6, 0); ρ (x, y) = x y$ .

Answer

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

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 $(0, 0), (1, 1)$ , and $(0, 5); ρ (x, y) = x + y$ .

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

Answer

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

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 $(0, 1), (0, 3), (3, 3)$ , and $(3, 1); ρ (x, y) = x^{2} y$ .

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

Answer

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

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 $y = 0, y = 1, y = x$ , and y = -x + 3; \space \rho (x,y) = 2x + y\).

7. [T] $R$ is the disk of radius $2$ centered at $(1, 2); ρ (x, y) = x^{2} + y^{2} - 2 x - 4 y + 5$ .

Answer

a. $M_{x} = 16 π, M_{y} = 8 π$ ; b. $\bar{x} = 1, \bar{y} = 2$ ;

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; $ρ (x, y) = 3 x^{4} + 6 x^{2} y^{2} + 3 y^{4}$ .

8. [T] $R$ is the region enclosed by the ellipse $x^{2} + 4 y^{2} = 1; ρ (x, y) = 1$ .

Answer

a. $M_{x} = 0, M_{y} = 0)$ ; b. $\bar{x} = 0, \bar{y} = 0$ ;

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

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

10. [T] $R$ is the region bounded by $y = x, y = - x, y = x + 2$ , and $y = - x + 2; ρ (x, y) = 1$ .

Answer

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

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 $y = \frac{1}{x}, y = \frac{2}{x}, y = 1$ , and $y = 2; ρ (x, y) = 4 (x + y)$ .

Exercise $3$

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ given in the first two groups of Exercises.

a. Find the moments of inertia $I_{x}, I_{y}$ 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 $(0, 0), (0, 3)$ , and $(6, 0); ρ (x, y) = x y$ .

Answer: a. $I_{x} = \frac{243}{10}, I_{y} = \frac{486}{5}$ , and $I_{0} = \frac{243}{2}$ ; b. $R_{x} = \frac{3 \sqrt{5}}{5}, R_{y} = \frac{6 \sqrt{5}}{5}$ , and $R_{0} = 3$

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

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

Answer: a. $I_{x} = \frac{2592 \sqrt{2}}{7}, I_{y} = \frac{648 \sqrt{2}}{7}$ , and $I_{0} = \frac{3240 \sqrt{2}}{7}$ ; b. $R_{x} = \frac{6 \sqrt{21}}{7}, R_{y} = \frac{3 \sqrt{21}}{7}$ , and $R_{0} = \frac{3 \sqrt{106}}{7}$

4. $R$ is the rectangular region with vertices $(0, 1), (0, 3), (3, 3)$ , and $(3, 1); ρ (x, y) = x^{2} y$ .

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

Answer: a. $I_{x} = 88, I_{y} = 1560$ , and $I_{0} = 1648$ ; b. $R_{x} = \frac{\sqrt{418}}{19}, R_{y} = \frac{\sqrt{7410}}{10}$ , and $R_{0} = \frac{2 \sqrt{1957}}{19}$

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

7. $R$ is the disk of radius $2$ centered at $(1, 2); ρ (x, y) = x^{2} + y^{2} - 2 x - 4 y + 5$ .

Answer: a. $I_{x} = \frac{128 π}{3}, I_{y} = \frac{56 π}{3}$ , and $I_{0} = \frac{184 π}{3}$ ; b. $R_{x} = \frac{4 \sqrt{3}}{3}, R_{y} = \frac{\sqrt{21}}{2}$ , and $R_{0} = \frac{\sqrt{69}}{3}$

8. $R$ is the unit disk; $ρ (x, y) = 3 x^{4} + 6 x^{2} y^{2} + 3 y^{4}$ .

9. $R$ is the region enclosed by the ellipse $x^{2} + 4 y^{2} = 1; ρ (x, y) = 1$ .

Answer: a. $I_{x} = \frac{π}{32}, I_{y} = \frac{π}{8}$ , and $I_{0} = \frac{5 π}{32}$ ; b. $R_{x} = \frac{1}{4}, R_{y} = \frac{1}{2}$ , and $R_{0} = \frac{\sqrt{5}}{4}$

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

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

Answer: a. $I_{x} = \frac{7}{3}, I_{y} = \frac{1}{3}$ , and $I_{0} = \frac{8}{3}$ ; b. $R_{x} = \frac{\sqrt{42}}{6}, R_{y} = \frac{\sqrt{6}}{6}$ , and $R_{0} = \frac{2 \sqrt{3}}{3}$

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

Exercise $4$ : Mass of a solid

1. Let $Q$ be the solid unit cube. Find the mass of the solid if its density $ρ$ is equal to the square of the distance of an arbitrary point of $Q$ to the $x y$ -plane.

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

2. Let $Q$ be the solid unit hemisphere. Find the mass of the solid if its density $ρ$ 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 $Q = {(x, y, z) | 1 \leq x^{2} + z^{2} \leq 25, y \leq 1 - x^{2} - z^{2}}$ whose density is $ρ (x, y, z) = k$ , where $k > 0$ .

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

a. Find the mass of $Q$ .

b. Find the moments $M_{x y}, M_{x z}$ and $M_{y z}$ about the $x y$ -plane, $x z$ -plane, and $y z$ -plane, respectively.

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

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

Answer

a. $m = \frac{9 π}{4}$ ; b. $M_{x y} = \frac{3 π}{2}, M_{x z} = \frac{81}{8}, M_{y z} = \frac{81}{8}$ ; c. $\bar{x} = \frac{9}{2 π}, \bar{y} = \frac{9}{2 π}, \bar{z} = \frac{2}{3}$ ;

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

a. Find the mass of $Q$ .

b. Find the moments $M_{x y}, M_{x z}$ and $M_{y z}$ about the $x y$ -plane, $x z$ -plane, and $y z$ -plane, respectively.

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

7. [T] The solid $Q$ has the mass given by the triple integral $\begin{matrix} (1) & \int_{- 1}^{1} \int_{0}^{π / 4} \int_{0}^{1} r^{2} d r d θ d z . \end{matrix}$

8. Use a CAS to answer the following questions.

10. The solid $Q$ is bounded by the planes $x + y + z = 3, x = 0, y = 0$ , and $z = 0$ . Its density is $ρ (x, y, z) = x + a y$ , 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 $ρ (x, y, z) = \frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}}$ , find $R$ such that the mass of the solid is $\frac{7 π}{2} .$

12. The mass of a solid $Q$ is given by $\begin{matrix} (2) & \int_{0}^{2} \int_{0}^{\sqrt{4 - x^{2}}} \int_{\sqrt{x^{2} + y^{2}}}^{\sqrt{16 - x^{2} - y^{2}}} {(x^{2} + y^{2} + z^{2})}^{n} d z d y d x, \end{matrix}$ where $n$ is an integer. Determine $n$ such the mass of the solid is $(2 - \sqrt{2}) π$ .

Answer: $n = - 2$

13. Let $Q$ be the solid bounded above the cone $x^{2} + y^{2} = z^{2}$ and below the sphere $x^{2} + y^{2} + z^{2} - 4 z = 0$ . 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 $Q = {(x, y, z) | 0 \leq x^{2} + y^{2} \leq 16, x \geq 0, y \geq 0, 0 \leq z \leq x}$ has the density $ρ (x, y, z) = k$ . Show that the moment $M_{x y}$ about the $x y$ -plane is half of the moment $M_{y z}$ about the $y z$ -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 $x y$ -plane, where $0 < a < b$ . Find the mass of the solid if its density is given by $ρ (x, y, z) = \sqrt{x^{2} + y^{2}}$ .

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 $M_{x y}$ about the $x y$ -plane is the same as the moment $M_{y z}$ about the $x z$ -plane.

17. The solid $Q$ has the mass given by the triple integral $\begin{matrix} (3) & \int_{0}^{1} \int_{0}^{π / 2} \int_{0}^{r^{3}} (r^{4} + r) d z d θ d r . \end{matrix}$

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

18. The solid $Q$ has the moment of inertia $I_{x}$ about the $y z$ -plane given by the triple integral $\begin{matrix} (4) & \int_{0}^{2} \int_{- \sqrt{4 - y^{2}}}^{\sqrt{4 - y^{2}}} \int_{\frac{1}{2} (x^{2} + y^{2})}^{\sqrt{x^{2} + y^{2}}} (y^{2} + z^{2}) (x^{2} + y^{2}) d z d x d y . \end{matrix}$

a. Find the density of $Q$ .

b. Find the moment of inertia $I_{z}$ about the $x y$ -plane.

Answer: a. $ρ (x, y, z) = x^{2} + y^{2}$ ; b. $\frac{16 π}{7}$

19. The solid $Q$ has the mass given by the triple integral $\begin{matrix} (5) & \int_{0}^{π / 4} \int_{0}^{2 s e c θ} \int_{0}^{1} (r^{3} c o s θ s i n θ + 2 r) d z d r d θ . \end{matrix}$

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

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

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

Answer: $M_{x y} = π (f (0) - f (a) + a f^{'} (a))$

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 $x y$ -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 $ρ (x, y, z) = f^{'} (y)$ , 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 $x z$ -plane of $Q$ is null.

23. [T] The average density of a solid $Q$ is defined as $\begin{matrix} (6) & ρ_{a v e} = \frac{1}{V (Q)} ∭_{Q} ρ (x, y, z) d V = \frac{m}{V (Q)}, \end{matrix}$ 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 $ρ (x, y, z) = e^{- x^{2} - y^{2} - z^{2}}$ , use a CAS to find its average density. Round your answer to three decimal places.

23. Show that the moments of inertia $I_{x}, I_{y}$ , and $I_{z}$ about the $y z$ -plane, $x z$ -plane, and $x y$ -plane, respectively, of the unit ball centered at the origin whose density is $ρ (x, y, z) = e^{- x^{2} - y^{2} - z^{2}}$ are the same. Round your answer to two decimal places.

Answer: $I_{x} = I_{y} = I_{z} \approx 0.84$

Search

Text Color

Text Size

Margin Size

Font Type

Exercise $1$ : Mass

Exercise $2$ (CAS)

Exercise $3$

Exercise $4$ : Mass of a solid

Exercise 1: Mass

Support Center

How can we help?

Exercise $1$ : Mass