다중적분의 응용(질량·질량중심)

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 $x\text{-axis}$ and $y\text{-axis,}$ 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 $R.$

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

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 $x\text{-axis}$ and $y\text{-axis,}$ 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 $R.$

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

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 $x\text{-axis}$ and $y\text{-axis,}$ 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 $R.$

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

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 $x\text{-axis}$ and $y\text{-axis,}$ 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 $R.$

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

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 ${I}_{x},{I}_{y},$ and ${I}_{0}$ about the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively. b. Find the radii of gyration with respect to the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively.

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

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 ${I}_{x},{I}_{y},$ and ${I}_{0}$ about the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively. b. Find the radii of gyration with respect to the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively.

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

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 ${I}_{x},{I}_{y},$ and ${I}_{0}$ about the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively. b. Find the radii of gyration with respect to the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively.

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

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 ${I}_{x},{I}_{y},$ and ${I}_{0}$ about the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively. b. Find the radii of gyration with respect to the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively.

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

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

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

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 ${I}_{x},{I}_{y},$ and ${I}_{0}$ about the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively. b. Find the radii of gyration with respect to the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively.

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

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 ${I}_{x},{I}_{y},$ and ${I}_{0}$ about the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively. b. Find the radii of gyration with respect to the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively.

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

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 ${I}_{x},{I}_{y},$ and ${I}_{0}$ about the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively. b. Find the radii of gyration with respect to the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively.

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

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 ${I}_{x},{I}_{y},$ and ${I}_{0}$ about the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively. b. Find the radii of gyration with respect to the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively.

$R=\{(x,y)|9{x}^{2}+{y}^{2}\le 1,x\ge 0,y\ge 0\};\rho (x,y)=\sqrt{9{x}^{2}+{y}^{2}}.$

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 ${I}_{x},{I}_{y},$ and ${I}_{0}$ about the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively. b. Find the radii of gyration with respect to the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively.

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

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 ${I}_{x},{I}_{y},$ and ${I}_{0}$ about the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively. b. Find the radii of gyration with respect to the $x\text{-axis},$ $y\text{-axis},$ and origin, respectively.

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

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 $xy\text{-plane}.$

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

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.

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