Answer
$$V = \frac{{500}}{3}\pi $$
Work Step by Step
$$\eqalign{
& y = \sqrt {50 - 2{x^2}} ,{\text{ }}x = 0{\text{ and }}y = 0 \cr
& y = \sqrt {50 - 2{x^2}} \to x = \sqrt {25 - \frac{1}{2}{y^2}} \cr
& {\text{Let }}\sqrt {25 - \frac{1}{2}{y^2}} = 0 \cr
& 25 - \frac{1}{2}{y^2} = 0 \cr
& {y^2} = 50 \cr
& y = \pm \sqrt {50} \cr
& {\text{Use the Shell method about the }}x{\text{ - axis}} \cr
& V = \int_c^d {2\pi y\left[ {p\left( y \right) - q\left( y \right)} \right]} dy \cr
& {\text{From the graph we can see that}} \cr
& \sqrt {25 - \frac{1}{2}{y^2}} > 0{\text{ on the interval }}\left[ {0,\sqrt {50} } \right] \cr
& {\text{Therefore}}{\text{,}} \cr
& V = \int_0^{\sqrt {50} } {2\pi y\left[ {\sqrt {25 - \frac{1}{2}{y^2}} } \right]} dy \cr
& V = - 2\pi \int_0^{\sqrt {50} } {\sqrt {25 - \frac{1}{2}{y^2}} \left( { - y} \right)} dy \cr
& {\text{Integrating}} \cr
& V = - 2\pi \left[ {\frac{{{{\left( {25 - \frac{1}{2}{y^2}} \right)}^{3/2}}}}{{3/2}}} \right]_0^{\sqrt {50} } \cr
& V = - \frac{4}{3}\pi \left[ {{{\left( {25 - \frac{1}{2}{y^2}} \right)}^{3/2}}} \right]_0^{\sqrt {50} } \cr
& V = - \frac{4}{3}\pi \left[ {{{\left( {25 - \frac{1}{2}{{\left( {\sqrt {50} } \right)}^2}} \right)}^{3/2}}} \right] + \frac{4}{3}\pi \left[ {{{\left( {25 - \frac{1}{2}{{\left( 0 \right)}^2}} \right)}^{3/2}}} \right] \cr
& {\text{Simplifying}} \cr
& V = - \frac{4}{3}\pi \left[ {{{\left( 0 \right)}^{3/2}}} \right] + \frac{4}{3}\pi \left[ {{{\left( {25} \right)}^{3/2}}} \right] \cr
& V = \frac{4}{3}\pi \left( {125} \right) \cr
& V = \frac{{500}}{3}\pi \cr} $$
