Answer
$\displaystyle{V=\frac{256\pi }{5}}\\ $
Work Step by Step
$\displaystyle{2x=y^2}\\
\displaystyle{x=\frac{y^2}{2}}$
$\displaystyle{A\left(y\right)=\pi \left(\frac{y^2}{2}\right)^2}\\
\displaystyle{A\left(y\right)=\pi \left(\frac{y^4}{4}\right)}\\$
$\displaystyle{V=\int_0^4A\left(y\right)\ dy}\\
\displaystyle{V=\int_0^4\pi \left(\frac{y^4}{4}\right)\ dy}\\
\displaystyle{V=\frac{\pi}{4}\int_0^4y^4\ dy}\\
\displaystyle{V=\frac{\pi}{4}\left[\frac{1}{5}y^5\right]_0^4}\\
\displaystyle{V=\frac{\pi}{4}\left(\left(\frac{1}{5}\times4^5\right)-\left(\frac{1}{5}\times0^5\right)\right)}\\
\displaystyle{V=\frac{256\pi }{5}}\\ $