Answer
$\dfrac{256 \pi}{15}$
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, $V=2\pi \int_{0}^{4} y \sqrt {4-y} \ dy\\= -2 \pi \int_0^4 (4-t) \sqrt t \ dt \\=-2 \pi [ \dfrac{8t^{3/2}}{3}-\dfrac{2y^{5/2}}{5}]_4^0 \\= -2\pi [\dfrac{-64}{3}+\dfrac{64}{5}] \\=\dfrac{256 \pi}{15}$
