Answer
$V = \frac{{2\pi }}{3}\left( {27 - 5\sqrt 5 } \right)$
Work Step by Step
$$\eqalign{
& y = \sqrt {5 + {x^2}} {\text{, }}y = 0,{\text{ }}x = 0,{\text{ }}x = 2 \cr
& {\text{Apply the shell method about the }}y{\text{ - axis}}{\text{.}} \cr
& V = \int_a^b {2\pi x} f\left( x \right)dx,{\text{ where }}0 \leqslant a \leqslant b \cr
& {\text{Therefore}}{\text{,}} \cr
& V = \int_0^2 {2\pi x\left( {\sqrt {5 + {x^2}} } \right)} dx \cr
& V = \pi \int_0^2 {2x{{\left( {5 + {x^2}} \right)}^{1/2}}dx} \cr
& {\text{Integrating}} \cr
& V = \pi \left[ {\frac{{{{\left( {5 + {x^2}} \right)}^{3/2}}}}{{3/2}}} \right]_0^2 \cr
& V = \frac{{2\pi }}{3}\left[ {{{\left( {5 + {x^2}} \right)}^{3/2}}} \right]_0^2 \cr
& {\text{Evaluating}} \cr
& V = \frac{{2\pi }}{3}\left[ {{{\left( {5 + {2^2}} \right)}^{3/2}} - {{\left( {5 + {0^2}} \right)}^{3/2}}} \right] \cr
& V = \frac{{2\pi }}{3}\left[ {{{\left( 9 \right)}^{3/2}} - {{\left( 5 \right)}^{3/2}}} \right] \cr
& V = \frac{{2\pi }}{3}\left( {27 - 5\sqrt 5 } \right) \cr} $$
