Answer
$\frac{384\pi}{7}$
Work Step by Step
Setup the integration using Shell method about the y-axis
$2\pi \int_0^4 x(8-x^{\frac{3}{2}} )dx$
$ 2\pi \int_0^4 (8x-x^{\frac{5}{2}} )dx$ , Integrate
$2\pi(4x^2 - \frac{2}{7}x^{\frac{7}{2}})]_0^4 $, Take the definite integral
$2\pi (64-\frac{256}{7})-0$
$ \frac{384\pi}{7}$
