Answer
$\displaystyle{V=8\pi}$
Work Step by Step
$\displaystyle{-x^2+6x-8=0}\\ \displaystyle{x^2-6x+8=0}\\ \displaystyle{x=2\qquad x=4}$
$\displaystyle{V=\int_{2}^{4}(2\pi x)\left(-x^2+6x-8\right)\ dx}\\ \displaystyle{V=2\pi\int_{2}^{4}6x^2-x^3-8x\ dx}\\ \displaystyle{V=2\pi\left[2x^3-\frac{1}{4}x^4-4x^2\right]_{2}^{4}}\\ \displaystyle{V=2\pi\left(\left(2(4)^3-\frac{1}{4}(4)^4-4(4)^2\right)-\left(2(2)^3-\frac{1}{4}(2)^4-4(2)^2\right)\right)}\\ \displaystyle{V=8\pi}$
