Answer
$64 \pi$
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, $V=2\pi \int_{-1}^{3} (4-y) (2y-y^2+3) \ dy \\= 2\pi \int_{-1}^{3} (8y-4y^2+12-2y^2+y^3-3y) \ dy \\= 2 \pi [\dfrac{5y^2}{2}-2y^3+12 y+\dfrac{y^4}{4}]_{-1}^3 \\=2 \pi (\dfrac{99}{4}+\dfrac{29}{4}) \\= 64 \pi$
