Answer
$\dfrac{40 \pi}{3}$
Work Step by Step
The shell method to compute the volume of a region: The volume of a solid obtained by rotating the region under $y=f(x)$ over an interval $[m,n]$ about the y-axis is given by:
$V=2 \pi \int_{m}^{n} (Radius) \times (height \ of \ the \ shell) \ dy=2 \pi \int_{m}^{n} (y) \times f(y) \ dy$
Now, $V=2\pi \int_{0}^{2} (2-x) (4-x^2) \ dx \\ = 2\pi [8x-\dfrac{2x^3}{3}-2x^2+\dfrac{x^4}{4}]_0^2 \\=2\pi [\dfrac{20}{3}] \\=\dfrac{40 \pi}{3}$
