Volumes of Revolution (College Board AP® Calculus BC): Exam Questions

Let $R$ be the region enclosed by the graphs of $f\left(x\right)=e^{\cos x}$ and $g\left(x\right)=\sqrt{x}-3$, the $y$-axis, and the vertical line $x=3$, as shown in the figure above.

Write, but do not evaluate, an integral expression that gives the volume of the solid generated when $R$ is rotated about the horizontal line $y=3$.

Let $R$ be the region in the first quadrant enclosed by the coordinate axes and the graph of $y={\left(\frac{x-2}{2}\right)}^2$, as shown in the figure above.

Find the volume of the solid generated when $R$ is rotated about the $x$-axis.

The inside of a vase of height $10$ inches has circular cross sections, as shown in the figure above. At height $h$, the radius of the vase is given by $r=\frac{1}{20}\left(8+h^2\right)$, where $0\le h\le 10$. The units of $r$ and $h$ are in inches.

Find the volume of the vase.

Let $R$ be the region in the first and second quadrants bounded above by the curve of $y=\frac{99}{1+2x^2}$ and below by the horizontal line $y=3$.

Find the volume of the solid generated when $R$ is rotated about the $x$-axis.

Let $R$ be the region in the first quadrant enclosed by the graphs $f\left(x\right)=\frac{3}{4}{x}^2$ and $g\left(x\right)=3\sin \left(\frac{\pi x}{4}\right)$ on the interval $0\le x\le 2$ as shown in the figure above.

Write, but do not evaluate, an integral expression for the volume of the solid generated when $R$ is rotated about the vertical line $x=2$.

A company designs fishing lures using the family of curves with equation ${\left(ky\right)}^2\left(9-y^2\right)-x^2=0$, where $k$ is a positive constant. The figure above shows the region in the first quadrant bounded by the $y$-axis and the curve ${\left(ky\right)}^2\left(9-y^2\right)-x^2=0$, for some $k$. Each fishing lure is in the shape of the solid generated when such a region is revolved about the $y$-axis. Both $x$ and $y$ are measured in inches.

For a particular fishing lure, the volume is $\frac{162\pi }{25}$ cubic inches. What is the value of $k$ for this fishing lure?

Let $R$ be the region in the first quadrant enclosed by the $y$-axis, the line $x=2$, and the graphs of $y=\sec \left(1-x\right)$ and $y=\frac{x^3}{8}$, as shown in the figure above.

Find the volume of the solid generated when $R$ is rotated about the $x$-axis.

Let $R$ be the region in the first quadrant enclosed by the coordinate axes, the line $y=8$, and the graphs of $y=x^2+1$ and $y={\left(x-4\right)}^3$, as shown in the figure above.

Find the volume of the solid generated when $R$ is rotated about the $y$-axis.

Let $f\left(x\right)=\frac{1}{2}{x}^3-4$ and $R$ be the region in the fourth quadrant bounded below by the graph $y=f\left(x\right)$, above by the $x$-axis and to the left by the $y$-axis.

Find the volume of the solid generated when $R$ is rotated about the horizontal line $y=-6$.

Let $f$ and $g$ be the functions given by $f\left(x\right)=\frac{3}{2}+\cos \left(2\pi x\right)$ and $g\left(x\right)=\ln \left(x+2\right)$. Let $R$ be the shaded region in the first quadrant enclosed by the $y$-axis and the graphs of $f$ and $g$, and let $S$ be the shaded region in the first quadrant enclosed by the graphs of $f$ and $g$ as shown in the figure above.

Find the volume of the solid generated when $R$ and $S$ are revolved around the horizontal line $y=-1$.