Answer
$18 \pi (2 \sqrt 2-1)$
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:
$V=2 \pi \int_{m}^{n} (Radius) \times (height \ of \ the \ shell) \ dy=2 \pi \int_{m}^{n} (x) \times f(x) \ dx$
Now, $V=2\pi \int_{0}^{3} (x)(\sqrt {x^2+9}) \ dx$
Let us use the substitution method. So, substitute $a=(x^2+9) \implies da=2x dx$
Now, $V= \pi \int_{9}^{18} a^{1/2} da \\=2 \pi [\dfrac{a^{3/2}}{3}]_{9}^{18} \\= 18 \pi (2 \sqrt 2-1)$
