Volumes of revolution Quiz

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

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

2. The curve $y=x^2$ for $0\leq y\leq4$ is rotated about the $y$-axis. Find the volume.

• A. $4\pi$
• B. $16\pi$
• C. $\frac{32\pi}{5}$
• D. $8\pi$

3. Which integral gives the volume when the region under $y=\mathrm{f}(x)$ between $x=a$ and $x=b$ is rotated about the $x$-axis?

• A. $\pi\int_a^by^2\,\mathrm{d}x$
• B. $2\pi\int_a^by\,\mathrm{d}x$
• C. $\pi\int_a^by\,\mathrm{d}x$
• D. $\pi\int_a^bx^2\,\mathrm{d}x$

4. The line $y=x$ for $0\leq x\leq3$ is rotated about the $x$-axis. Find the volume of the cone formed.

• A. $3\pi$
• B. $6\pi$
• C. $9\pi$
• D. $27\pi$

5. (A2) A curve has $x=t^2$ and $y=2t$ for $0\leq t\leq2$. It is rotated about the $x$-axis. Find the volume.

• A. $16\pi$
• B. $32\pi$
• C. $64\pi$
• D. $8\pi$