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, $V=2\pi \int_{2}^{6} (6-y) \sqrt {y-2} \ dy$
Let us use substitution method. So, substitute $a=y-2 \implies da= dy$
$V= 2 \pi \int_0^4 (4-a) a^{1/2} \ dy \\= 2\pi [\dfrac{8}{3}a^{3/2}-\dfrac{2}{5}a^{5/2}]_0^4 \\=2\pi [\dfrac{8}{3}(y-2)^{3/2}-\dfrac{2}{5}(y-2)^{5/2}]_0^4 \\=\dfrac{256 \pi}{15}$
