Volume of Revolution Quiz (Disk, Washer, Shell Methods)

Question 1

Find the volume of the solid formed by rotating the region under $y=x^2$ from $x=0$ to $x=2$ about the $x$-axis.

Accepted Answer

$\dfrac{32\pi}{5}$

Answer

$\dfrac{8\pi}{3}$

Answer

$\dfrac{16\pi}{5}$

Answer

$\dfrac{64\pi}{5}$

Question 2

Using washers, find the volume when the region between $x=y^2$ and $x=4$ is revolved about the $y$-axis (for $-2\le y\le 2$).

Accepted Answer

$\dfrac{128\pi}{3}$

Answer

$\dfrac{64\pi}{3}$

Answer

$\dfrac{384\pi}{5}$

Answer

$32\pi$

Question 3

A region is bounded by $y=x$ and $y=\dfrac{1}{x}$ for $1\le x\le 2$. It is revolved about the $y$-axis. Which expression gives the volume with the fewest changes of variable, and what is the correct setup?

Accepted Answer

Shell method: $V=2\pi\int_{1}^{2} x\left(x-\dfrac{1}{x}\right)\,dx$

Answer

Shell method: $V=\pi\int_{1}^{2} x\left(x-\dfrac{1}{x}\right)\,dx$

Answer

Disk method: $V=\pi\int_{1}^{2} (x-\dfrac{1}{x})^2\,dx$

Answer

Washer method: $V=\pi\int_{1}^{2} \left(x^2-\dfrac{1}{x^2}\right)\,dx$

Volume of Revolution Quiz (Disk, Washer, Shell Methods)

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