Answer
$94.78$
Work Step by Step
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, $V=2\pi \int_{0.585}^{3.415} y (8y-2y^2-4y) \ dy\\= 2 \pi [\dfrac{8y^3}{3}-\dfrac{2y^4}{4}-\dfrac{4y^2}{2}]_{0.585}^{3.415} \\= 2\pi [14.8758+0.2091] \\=94.78$
