Answer
$\dfrac{3200 \pi }{3}$
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, $V=2\pi \int_{0}^{8} (x+3) (2x) \ dx \\= 2\pi \int_{0}^{8} (2x^2+6x) \ dx \\= 2 \pi [\dfrac{2x^3}{3}+3x^2]_{0}^{8} \\=2 \pi [\dfrac{2(8^3)}{3}+(3) (64)] \\= \dfrac{3200 \pi }{3}$
