Answer
$\dfrac{128 \pi}{5}$
Work Step by Step
Shell method to compute the volume of a region: The volume of a solid obtained by rotating the region under $y=f(x)$ over the 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, $V=2\pi \int_0^{4} (x)(x^{1/2}) \ dx\\=2\pi \int_0^{1} (x^{3/2}) \ dx \\=2 \pi [\dfrac{x^{5/2}}{5/2}]_0^{4} \\=\dfrac{128 \pi}{5}$
