If a function $f$ is continuous on a closed interval $[a,b]$, then the volume $V$ of the solid of revolution obtained by revolving the region bounded by the graph of $f$, the $x$-axis, and the lines $x = a$ and $x = b$ about the $x$-axis, is found using the formula $V =$_____.

True or False When the region bounded by the graphs of the functions $f$ and $g$ and the lines $x=a$ and $x=b$ is revolved about the $x$-axis, the cross section exposed by making a slice at $u_i$ perpendicular to the $x$-axis is two concentric circles, and the area $A_{i}$ between the circles is $A_{i}=\pi [f(u_{i})-g(u_{i})]^{2}$.

True or False If the functions $f$ and $g$ are continuous on the closed interval $[a,b]$ and if $f(x) \geq g(x) \geq 0$ on the interval, then the volume $V$ of the solid of revolution obtained by revolving the region bounded by the graphs of $f$ and $g$ and the lines $x = a$ and $x = b$ about the $x$-axis is $V=\pi \int^b_a [f(x) - g(x)]^2~{\it dx}$.

True or False If the region bounded by the graphs of $y=x^{2}$ and $y=2x$ is revolved about the line $y=6$, the volume $V$ of the solid of revolution generated is found by finding the integral $$V=\pi \int_{6}^{7}( 4x^2-x^{4})\, {\it dx}.$$

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

$y=x^{4}$ about the $y$-axis

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

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

$y=\sec x$ about the $x$-axis

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

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

$y=\sqrt{x}$, the $x$-axis, $x=4, x=9$; about the $x$-axis

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

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

$y=x^{2}, x\geq 0, y=1, y=4$; about the $y$-axis

$y=2\sqrt{x}$, the $y$-axis, $y=4$; about the $y$-axis

$y=x^{2}, x\geq 0$, the $y$-axis, $y=4$; about the $x$-axis

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

$y=2\sqrt{x}$, the $y$-axis, $y=4$; about the $x$-axis

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

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

$y=2x^{4}$, the $x$-axis, $x=1$; about the $y$-axis

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

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

$y=\sqrt{x}$, the $y$-axis, $y=9$; about the $y$-axis

$y=\sqrt{x}$, the $y$-axis, $y=9$; about the $x$-axis

$y=(x-2) ^{3}$, the $x$-axis, $x=0, x=3$; about the $x$-axis

$y=(x-2) ^{3}$, the $x$-axis, $x=0, x=3$; about the $y$-axis

$y=(x+1) ^{2}, x\geq 0, y=16$; about the $y$-axis

$y=(x+1) ^{2}, x\leq 0, y=16$; about the $x$-axis

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

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

$y=4x, \ y=x^{3}, x\geq 0;$ about the $x$-axis

$y=2x+1, y=x, x=0, x=3;$ about the $x$-axis

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

$y=\cos x, y=\sin x, x=0, x=\dfrac{\pi}{4};$ about the $x$-axis

$y=\csc x, y=0, x= \dfrac{\pi}{2}, x=\dfrac{3\pi }{4};$ about the $x$-axis

$y=\sec x, y=0, x=0, x=\dfrac{\pi}{3}$; about the $x$-axis

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

$y=\dfrac{1}{x}, y=0, x=1, x=4$; about $y=4$

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

$y=x^{3}, x=0, y=1$; about $x=-1$

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

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

$y=\dfrac{1}{x^{2}}, y=0, x=1, x=4$; about $y=4$

$y=\sqrt{x}, y=0, 0\leq x\leq 4$; about $y=-4$

Volume of a Solid of Revolution A region in the first quadrant is bounded by the $x$-axis and the graph of $y=kx-x^{2}$, where $k > 0$.

Volume of a Solid of Revolution

Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of $y=\cos x$ and $y=0$ from $x=0$ to $x=\dfrac{\pi}{2}$ about the line $y=1$. $\left[ \hbox{Hint: }\cos ^{2}x=\dfrac{1+\cos (2x) }{2}.\right]$

Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of $y=\cos x$ and $y=0$ from $x=0$ to $x=\dfrac{\pi}{2}$ about the line $y=-1$. (See the hint in Problem 49.)

Volume of a Solid of Revolution The graph of the function $P(x)=kx^{2}$ is symmetric with respect to the $y$-axis and contains the points $(0,0)$ and $(b,e^{-b^{2}})$, where $b > 0$.

Volume of a Solid of Revolution Find the volume of the solid generated by revolving the region bounded by the catenary $y=a~\cosh \left( \dfrac{x}{a}\right) +b-a$, the $x$-axis, $x=0$, and $x=1$ about the $x$-axis, where $a > 0$ and $b \geq 0$. $\bigg[\textit{Hint:} \cosh ^{2}~x=\dfrac{\cosh ( 2x) +1}{2}.\bigg]$