Volumes of revolution and mean value Quiz

1. The region under $y=\sqrt{x}$ for $0\le x\le4$ is rotated through $2\pi$ about the $x$-axis. Find the volume.

• A. $4\pi$
• B. $8\pi$
• C. $16\pi$
• D. $\dfrac{32\pi}{3}$

2. Find the mean value of $\mathrm{f}(x)=x^2$ over $0\le x\le3$.

• A. $9$
• B. $\dfrac92$
• C. $6$
• D. $3$

3. The line $y=3x$ for $0\le x\le1$ is rotated through $2\pi$ about the $x$-axis. Find the volume.

• A. $3\pi$
• B. $9\pi$
• C. $\pi$
• D. $\dfrac{9\pi}{2}$

4. The curve $x=y^2$ for $0\le y\le2$ is rotated through $2\pi$ about the $y$-axis (region between the curve and the $y$-axis). Find the volume.

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

5. Find the mean value of $\sin x$ over $0\le x\le\pi$.

• A. $\dfrac1\pi$
• B. $\dfrac2\pi$
• C. $2$
• D. $\dfrac12$