Answer
$\frac{128\sqrt 2 \pi}{15}$
Work Step by Step
Setup the integration using shell method about the y-axis
$ 2\pi \int_2^4 x(\sqrt {x-2})dx$
$2\pi \int_2^4 (u+2)(u^{\frac{1}{2}})du$ , Use u-substitution
$ 2\pi \int_2^4 (u^{\frac{3}{2}} +2u^{\frac{1}{2}})du$
$2\pi (\frac{2}{5}u^{\frac{5}{2}} +\frac{4}{3} u^{\frac{3}{2}})]_0^2$, use change of variables
$2\pi (\frac{8\sqrt 2}{5} + \frac{8\sqrt 2}{3})$
$\frac{128\sqrt 2 \pi}{15}$
