Answer
$\displaystyle{V=\frac{136\pi}{15}}$
Work Step by Step
$\displaystyle{\sqrt x=\frac{x}{2}}\\
\displaystyle{2\sqrt x-x=0}\\
\displaystyle{x^{\frac{1}{2}}\left(2-x^{\frac{1}{2}}\right)=0}\\
\displaystyle{x=0\qquad x=4}$
$\displaystyle{V=\int_{0}^{4}(2\pi (5-x))\left(x^{\frac{1}{2}}-\frac{x}{2}\right)\ dx}\\
\displaystyle{V=2\pi\int_{0}^{4}5x^{\frac{1}{2}}-\frac{5}{2}x-x^{\frac{3}{2}}+\frac{1}{2}x^2\ dx}\\
\displaystyle{V=2\pi\left[\frac{10}{3}x^{\frac{3}{2}}-\frac{5}{4}x^2-\frac{2}{5}x^{\frac{5}{2}}+\frac{1}{6}x^{3}\right]_{0}^{4}}\\
\displaystyle{V=2\pi\left(\left(\frac{10}{3}(4)^{\frac{3}{2}}-\frac{5}{4}(4)^2-\frac{2}{5}(4)^{\frac{5}{2}}+\frac{1}{6}(4)^{3}\right)-\left(0\right)\right)}\\
\displaystyle{V=\frac{136\pi}{15}}$
