Answer
$V = \frac{{128\pi }}{5}$
Work Step by Step
$$\eqalign{
& y = \sqrt x ,{\text{ }}y = 0,{\text{ }}x = 4 \cr
& {\text{Apply the shell method about the }}y{\text{ - axis}}{\text{.}} \cr
& V = \int_a^b {2\pi x} f\left( x \right)dx,{\text{ where }}0 \leqslant a \leqslant b \cr
& {\text{Therefore}}{\text{,}} \cr
& V = \int_0^4 {2\pi x\left( {\sqrt x } \right)} dx \cr
& V = 2\pi \int_0^4 {{x^{3/2}}} dx \cr
& {\text{Integrating}} \cr
& V = 2\pi \left[ {\frac{{{x^{5/2}}}}{{5/2}}} \right]_0^4 \cr
& V = \frac{{4\pi }}{5}\left[ {{x^{5/2}}} \right]_0^4 \cr
& {\text{Evaluating}} \cr
& V = \frac{{4\pi }}{5}\left[ {{{\left( 4 \right)}^{5/2}} - {{\left( 0 \right)}^{5/2}}} \right] \cr
& V = \frac{{4\pi }}{5}\left( {32 - 0} \right) \cr
& V = \frac{{128\pi }}{5} \cr} $$
