Answer
$\frac{16\pi}{3}$
Work Step by Step
Setup the integration using shell method about the y-axis.
Setup the integration using Shell Method
$ 2\pi \int_0^2 x((4x-x^2) - x^2)dx$
$ 2 \pi \int_0^2 (4x^2 - 2x^3)dx $ ,Bring a factor of 2 out
$ 4\pi \int_0^2 (2x^2 - x^3)dx$
$ 4\pi (\frac{2}{3} x^3 - \frac{1}{4} x^4)]_0^2 $, Integrate
$ 4\pi ( \frac{16}{3} -4) -0 $, Take Definite Integral
$\frac{16\pi}{3}$
