Answer
$\dfrac{776\pi}{15}$
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, $V=2\pi \int_{0}^{2} (x^2+4)^2 \ dx\\= 2 \pi \int_0^2 (x^4+16+8x^2-4) \ dx \\= \pi [\dfrac{x^5}{5}+12x+\dfrac{8x^3}{3}]_0^2 \\=2 \pi [\dfrac{32}{5}+24+\dfrac{63}{3}] \\=\dfrac{776\pi}{15}$
