If a lamina represented by a closed, bounded region $R$ has a continuous mass density function $\rho =\rho (x, y) ,$ then the double integral $\iint\limits_{\kern-3ptR}\rho (x, y)\,{\it dA}$ gives the _____ of the lamina.

True or False If a lamina represented by a closed, bounded region $R$ has a continuous mass density function $\rho =\rho (x, y) ,$ then $\iint\limits_{\kern-3ptR}y\,\rho (x, y)\,{\it dA}$ is called the moment of mass about the $y$-axis.

If a lamina represented by a closed, bounded region $R$ has a continuous mass density function $\rho =\rho (x, y) ,$ the moment of mass about the $y$-axis $M_{y}$ is given by the double integral _____.

True or False Center of mass and centroid are synonyms.

True or False The moment of mass $M_a$ of a particle about the $a$-axis measures the tendency of the particle to rotate about the $a$-axis.

True or False The moment of inertia about the origin equals the difference between the moment of inertia about the $x$-axis and the moment of inertia about the $y$-axis.

Find the mass $M$ and center of mass $(\bar{x}, \bar{y})$ of the triangular lamina shown in the figure. The mass density function of the lamina is $\rho =\rho (x, y) =x^{2}y$.

Find the mass $M$ and the center of mass $(\bar{x}, \bar{y})$ of the square lamina shown in the figure. The mass density function of the lamina is $\rho =\rho (x, y) =xy^{2}$.

The lamina enclosed by the lines $x=2$ and $y=4$, and the coordinate axes, and whose mass density is $\rho =\rho (x, y) =3x^{2}y$

The lamina enclosed by the lines $x=1$ and $y=2$, and the coordinate axes, and whose mass density is $\rho =\rho (x, y) =2x^{2}y^{2}$

The lamina in the first quadrant enclosed by $y^{2}=x$, $x=1$, and the $x$-axis, and whose mass density is $\rho =\rho (x, y) =2x+3y$

The lamina in the first quadrant enclosed by $y^{2}=4x$, $\ x=1$, and the $x$-axis, and whose mass density is $\rho =\rho (x, y) =x+1$

The lamina enclosed by $y^{2}=x$ and $y=x,$ and whose mass density $\rho =\rho (x, y)$ is proportional to the distance from the $y$-axis

The lamina enclosed by $y=\sin x$ and the $x$-axis, $0\leq$ $x\leq \pi$, and whose mass density $\rho =\rho (x, y)$ is proportional to the distance from the $x$-axis

The lamina enclosed by the lines $2x+3y=6$, $x=0$, and $y=0,$ and whose mass density $\rho =\rho (x, y)$ is proportional to the sum of the distances from the coordinate axes

The lamina enclosed by the lines $3x+4y=12$, $x=0$, and $y=0,$ and whose mass density $\rho =\rho (x, y)$ is proportional to the product of the distances from the coordinate axes

The lamina enclosed by the cardioid $r=1+\sin \theta,$ and whose mass density $\rho =\rho (r, \theta)$ is proportional to the distance from the pole

The lamina enclosed by $r=\cos \theta ,$ $-\dfrac{\pi}{2}\leq \theta \leq \dfrac{\pi}{2}$, whose mass density $\rho =\rho (r,\theta)$ is proportional to the distance from the pole

A lamina inside the graph of $r=2a\sin \theta$ and outside the graph of $r=a,$ whose mass density $\rho =\rho (r, \theta)$ is inversely proportional to the distance from the pole

A lamina outside the limaçon $r=2-\cos \theta$ and inside the circle $r=4\cos \theta$, whose mass density $\rho =\rho (r,\theta)$ is inversely proportional to the distance from the pole

The lamina enclosed by the lines $2x+3y=6$, $x=0$, and $y=0$ about the $x$-axis

Rework Problem 21 for the moment of inertia about the $y$-axis.

The lamina in the first quadrant enclosed by the lines $x=a$ and $y=b$ about the $y$-axis

Rework Problem 23 for the moment of inertia about the $x$-axis.

The lamina enclosed by $y=x^{2}$ and $y=2-x^{2}$; about the $y$-axis

The lamina enclosed by the loop of $y^{2}=x^{2}(4-x)$; about the $y$-axis

Mass Find the mass of a flat circular washer with inner radius $a$ and outer radius $2a$ if its mass density $\rho =\rho (x, y)$ is inversely proportional to the square of the distance from the center. See the figure.

Mass Rework Problem 27 if the mass density $\rho =\rho (x, y)$ is inversely proportional to the distance from the center.

Mass and Center of Mass Find the mass and center of mass of a lamina in the shape of the region enclosed on the left by the line $x=a,$ $a\gt0$, and on the right by the circle $r=2a\cos \theta$ if its mass density $\rho$ is inversely proportional to the distance from the $y$-axis.

Mass and Center of Mass Find the mass and center of mass of a lamina in the shape of the smaller region cut from the circle $r=6$ by the line $r\cos \theta =3$ if its mass density is $\rho =\rho (r,\theta) =\cos ^{2}\theta$.

Mass Find the mass of the lamina enclosed by $y=x^{2}$ and $y=x^{3}$; the mass density is $\rho =\rho (x,y) =\sqrt{xy}$.

Center of Mass Find the center of mass of the lamina inside $r=4\cos \theta$ and outside $r=2\sqrt{3}$ if the mass density $\rho =\rho (r, \theta)$ is inversely proportional to the distance from the origin.

Center of Mass Find the mass and the center of mass of the lamina enclosed by $x=y-2$ and $x=-y^{2},$ if the mass density is $\rho =\rho (x,y) =x^{2}$.

Center of Mass Find the center of mass of the lamina enclosed by $y=x^{2}$ and $x-2y+1=0,$ if the mass density is $\rho =\rho (x,y) =2x+8y+2$.

Center of Mass Find the center of mass of the homogeneous lamina enclosed by $bx^{2}=a^{2}y$ and $ay=bx$, $a \gt 0, b \gt 0$.

Center of Mass Find the center of mass of the homogeneous lamina enclosed by $y=\ln x,$ the $x$-axis, and the line $x=e^{2}$.

Moment of Inertia The mass density at each point of a circular washer of inner radius $a$ and outer radius $b$ is inversely proportional to the square of the distance from the center.

Moment of Inertia

Show that the center of mass of a rectangular homogeneous lamina lies at the intersection of its diagonals.

A homogeneous lamina is in the shape of the region enclosed by a right triangle of base $b$ and height $h$. Show that its moment of inertia about the base is $\dfrac{1}{6}mh^{2}$, where $m$ is the mass of the lamina. (Hint: Position the triangle so that the base lies on the positive $x$-axis from $(0, 0)$ to ($b, 0$).)

Show that the center of mass of the region enclosed by a triangular homogeneous lamina lies at the point of intersection of its medians. (Hint: Position the triangle so that its vertices are at ($a, 0$), $(b, 0)$, and ($0, c$), where $a \lt 0$, $b \gt 0$, and $c \gt 0$)

Suppose that a homogeneous lamina occupies a region $R$ in the $xy$-plane. Show that the average value of $f(x, y)=x$ over $R$ is $\bar{x}$, the first coordinate of the center of mass of the lamina. What is the average value of $g(x, y)=y$ over $R$? [Hint: The average value of $f$ over a region $R$ is defined to be the number $\dfrac{1 }{A}\iint\limits_{\kern-3ptR}f(x, y)\,{\it dA}$.]