Answer
$\frac{16\pi}{3}$
Work Step by Step
Setup the integration using shell method about the x-axis
$ 2\pi \int _0^2 (y(4-y-y))dy$
$ 2\pi \int_0^2(4y-2y^2)dy$
$ 4\pi \int_0^2 (2y-y^2)dy$
$ 4\pi[y^2 - \frac{1}{3} y^3]_0^2$
$4\pi (4-\frac{8}{3})$
$\frac{16\pi}{3}$
