Answer
$\displaystyle{V=\frac{384\pi }{5}}\\ $
Work Step by Step
$\displaystyle{6-x^2=2}\\ \displaystyle{x^2=4}\\ \displaystyle{x=2 \qquad x=-2}\\$
$\displaystyle{A\left(x\right)=\pi \left(6-x^2\right)^2-\pi\left(2\right)^2}\\ \displaystyle{A\left(x\right)=\pi \left(x^4-12x+32\right)}\\$
$\displaystyle{V=\int_{-2}^2A\left(x\right)\ dx}\\ \displaystyle{V=\int_{-2}^2\pi \left(x^4-12x+32\right)\ dx}\\ \displaystyle{V=\pi \int_{-2}^2\left(x^4-12x+32\right)\ dx}\\ \displaystyle{V=\pi\left[\frac{1}{5}x^5-6x^2+ 32x\right]_{-2}^2}\\ \displaystyle{V=\pi\left(\left(\frac{1}{5}\times2^5-6\times 2^2+ 32\times2\right)-\left(\frac{1}{5}\times{-2}^5-6\times {-2}^2+ 32\times-2\right)\right)}\\ \displaystyle{V=\frac{384\pi }{5}}\\ $