Answer
$=\frac{162}{5}$
Work Step by Step
Setup the integration using shell method about the line x=3
$2\pi\int_{0}^3 (3-x)(6x-x^2-\frac{1}{3}x^3)$
$=2\pi\int_{0}^3 (18x-3x^2-x^3-6x^2+x^3+\frac{1}{3}x^4)$
$=2\pi\int_{0}^3 (\frac{1}{3}x^4-9x^2+18x)$
$=2\pi[\frac{x^5}{15}-3x^3+9x^2]_0^3$
$=\frac{162\pi}{5}$
