Answer
$\displaystyle{V=\frac{64\pi }{15}}\\ $
Work Step by Step
$y^2=2y\\ y^2-2y=0\\ y(y-2)=0\\ y=0 \qquad y=2$
$\displaystyle{A\left(y\right)=\pi \left(2y\right)^2-\pi \left(y^2\right)^2}\\ \displaystyle{A\left(y\right)=\pi \left(4y^2-y^4\right)}\\$
$\displaystyle{V=\int_0^2A\left(y\right)\ dy}\\ \displaystyle{V=\int_0^2\pi \left(4y^2-y^4\right)\ dy}\\ \displaystyle{V=\pi \int_0^24y^2-y^4\ dy}\\ \displaystyle{V=\pi\left[\frac{4}{3}y^3-\frac{1}{5}y^5\right]_0^2}\\ \displaystyle{V=\pi\left(\left(\frac{4}{3}2^3-\frac{1}{5}2^5\right)-\left(\frac{4}{3}0^3-\frac{1}{5}0^5\right)\right)}\\ \displaystyle{V=\frac{64\pi }{15}}\\ $
