Volumes of revolution Quiz

1. The region under $y=x^2$ from $x=0$ to $x=2$ is rotated about the $x$-axis. Find the volume.

• A. $\frac{8\pi}{3}$
• B. $16\pi$
• C. $\frac{32\pi}{5}$
• D. $\frac{16\pi}{5}$

2. The region under $y=\frac1x$ from $x=1$ to $x=2$ is rotated about the $x$-axis. Find the volume.

• A. $\frac{\pi}{2}$
• B. $\pi\ln2$
• C. $\frac{3\pi}{2}$
• D. $\frac{\pi}{4}$

3. A curve has parametric equations $x=f(t)$, $y=g(t)$. Which gives the volume when rotated about the $x$-axis, from $t=\alpha$ to $t=\beta$?

• A. $\pi\int_\alpha^\beta y\,dt$
• B. $\pi\int_\alpha^\beta y^2\,dt$
• C. $\pi\int_\alpha^\beta y\frac{dx}{dt}\,dt$
• D. $\pi\int_\alpha^\beta y^2\frac{dx}{dt}\,dt$

4. The region under $y=\sin x$ from $x=0$ to $x=\pi$ is rotated about the $x$-axis. Find the volume.

• A. $\pi^2$
• B. $\frac{\pi^2}{2}$
• C. $\frac{\pi}{2}$
• D. $2\pi$

5. A curve has $x=2t$, $y=t^2$ for $0\le t\le1$. It is rotated about the $x$-axis. Find the volume.

• A. $\frac{\pi}{5}$
• B. $\frac{4\pi}{5}$
• C. $\frac{2\pi}{5}$
• D. $\frac{\pi}{10}$