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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.

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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.

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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 _____.

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True or False Center of mass and centroid are synonyms.

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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.

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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.

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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 \(\rho =\rho (x, y) =x^{2}y\).

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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 \(\rho =\rho (x, y) =xy^{2}\).

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

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

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

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

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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

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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

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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

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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

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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

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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

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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

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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

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

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Rework Problem 21 for the moment of inertia about the \(y\)-axis.

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

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Rework Problem 23 for the moment of inertia about the \(x\)-axis.

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

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

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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.

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

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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.

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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 \(\rho =\rho (r,\theta) =\cos ^{2}\theta \).

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

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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.

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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 \( \rho =\rho (x,y) =x^{2}\).

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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 \(\rho =\rho (x,y) =2x+8y+2\).

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

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

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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.

1. Find the mass of the washer. Then discuss its behavior as \( a\rightarrow 0^+\). (Note that the mass density becomes unbounded as we approach the center of the washer.)
2. Find the moment of inertia of the washer about its center, and show that (unlike the mass) it remains finite as \(a\rightarrow 0^+\).

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Moment of Inertia

1. Graph the limaçon \( r=3+2\cos \theta \) and the circle \(r=2.\)
2. Set up the integral for the moment of inertia about the \(x\)-axis of the lamina inside the limaçon and outside the circle from (a) if the mass density \(\rho \) of the lamina is inversely proportional to the square of the distance from the origin.
3. Find the integral in (b).

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

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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 (\(b, 0\)).)

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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 \(c \gt 0\))

Question

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 \(\dfrac{1 }{A}\iint\limits_{\kern-3ptR}f(x, y)\,{\it dA}\).]

Question

1. Show that the moment of inertia about the \(z\)-axis of the thin flat plate in the figure equals the sum of its moments of inertia about the \(x\)- and \(y\)-axes.
   

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2. Given that the moment of inertia of a homogeneous disk about an axis through its center and perpendicular to its plane is \(\dfrac{mR^{2}}{2}\) (where \(m\) is mass and \(R\) is the radius), use (a) to find its moment of inertia about a diameter.
3. What is the moment of inertia of a disk about an axis tangent to its edge?