True or False In using the shell method to find the volume of a solid revolved about the $x$-axis, the integration occurs with respect to $x$.

True or False The volume of a cylindrical shell of outer radius $R$, inner radius $r$, and height $h$ is given by $V=\pi r^{2}h$.

True or False The volume of a solid of revolution can be found using either washers or cylindrical shells only if the region is revolved about the $y$-axis.

True or False If $y=f(x)$ is a function that is continuous and nonnegative on the closed interval $[a, b], a\geq 0$, then the volume $V$ of the solid generated by revolving the region bounded by the graph of $f$ and the $x$-axis from $x=a$ to $x=b$ about the $y$-axis is $V=2\pi \int_{a}^{b}xf(x)~{\it dx}$.

$y=x^{2}+1$, the $x$-axis, $0\leq x\leq 1$; about the $y$-axis

$y=x^{3}, y=x^{2}$; about the $y$-axis

$y=\sqrt{x}, y=x^{2}$; about the $y$-axis

$y=\dfrac{1}{x}$, the $x$-axis, $x=1, x=4$; about the $y$-axis

$y=x^{3}$, the $y$-axis, $y=8$; about the $x$-axis

$y=\sqrt{x}$, the $y$-axis, $y=2$; about the $x$-axis

$x=\sqrt{y}$, the $y$-axis, $y=1$; about the $x$-axis

$x=4\sqrt{y}$, the $y$-axis, $y=4$; about the $x$-axis

$y=x, y=x^{2}$; about the $x$-axis

$y=x, y=x^{3}$; in the first quadrant; about the $x$-axis

$y=e^{-x^{2}}$, and the $x$-axis, from $x=0$ to $x=2$; about the $y$-axis

$y=x^{3}+x$ and the $x$-axis, $0\leq x\leq 1;$ about the $y$-axis

$y=x^{2}, y=4x-x^{2}$; about $x=4$

$y=x^{2}, y=4x-x^{2}$; about $x=3$

$y=x^{2}, y=0, x=1, x=2$; about $x=1$

$y=x^{2}, y=0, x=1, x=2$; about $x=-2$

$x=y-y^{2}$, the $y$-axis; about $y=-1$

$x=y-y^{2}$, the $y$-axis; about $y=1$

$y=\sqrt[3]{x}$, the $y$-axis, $y=2$; about the $x$-axis

$y=\sqrt{x}+x$, the $x$-axis, $x=1, x=4$; about the $y$-axis

$y=x^{3}$ and $y=x$ to the right of $x=0$; about the $y$-axis

$y=x^{3}, y=x^{2}$; about the $x$-axis

$y=3x^{2}$ and $y=30-x$ to the right of $x=1$; about the $y$-axis

$y=3x^{2}$ and $y=30-x$ to the right of $x=1$; about the $x$-axis

$y=x^{2}$ and $y=8-x^{2}$ to the right of $x=1$; about the $x$-axis

$y=x^{2}$ and $y=8-x^{2}$ to the right of $x=1$; about the $y$-axis

$y=x^{2}, y=x$; about the $y$-axis

$y=x^{2/3}+x^{1/3}$, the $x$-axis, $x=1, x=8$; about the $y$-axis

$y=\sqrt{x}$ and $y=18-x^{2}$ to the right of $x=1$; about the $y$-axis

$y=\sqrt{x}$ and $y=18-x^{2}$ to the right of $x=1$; about the $x$-axis

Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graph of $y=\dfrac{1}{(x^2+1)^2}$ and the $x$-axis from $x=0$ to $x=1$ about the $y$-axis as shown in the figure.

Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graph of $y=\sqrt{x+1}+x$ and the $x$-axis from $x=0$ to $x=3$ about the $y$-axis.

Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region in the first quadrant bounded by the $x$-axis and the graph of $y=2x-x^{2}$:

Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region in the first quadrant bounded by the $x$-axis and the graph of $y=x\sqrt{9-x}$.

Volume of a Solid of Revolution Suppose $f(x)\geq 0$ for $x\geq 0$, and the region bounded by the graph of $f$ and the $x$-axis from $x=0$ to $x=k, k > 0$, is revolved about the $x$-axis. If the volume of the resulting solid is $\dfrac{1}{5}k^{5}+k^{4}+\dfrac{4}{3}k^{3}$, find $f$.

Volume of a Solid of Revolution Find the volume of the solid generated by revolving about the $y$-axis the region bounded by the graph of $y=e^{-x^{2}}$ and the $y$-axis from $x=0$ to the positive $x$-coordinate of the point of inflection of $y=e^{-x^{2}}$.

Volume of a Solid of Revolution Show that the volume $V$ of the solid generated by revolving the region bounded by the graph of $y=\sin ^{3}x$ and the $x$-axis from $x=0$ to $x=\pi$ about the line $x=-\pi$ is given by $V=4\pi^{2}$.

Comparing Volume Formulas In the figure below, the region bounded by the graphs of $y=g(x), y=f(x)$, and the $y$-axis is to be revolved about the $y$-axis. Show that the resulting volume $V$ is given by:

Suppose a plane region of area $A$ that lies to the right of the $y$-axis is revolved about the $y$-axis, generating a solid of volume $V$. If this same region is revolved about the line $x=-k, k > 0$, show that the solid generated has volume $V+2\pi kA$.

Find the volume generated by revolving the region bounded by the parabola $y^{2}=8x$ and its latus rectum about the latus rectum:

(The latus rectum is the chord through the focus perpendicular to the axis of the parabola.) See the figure.