Answer
$$V = \frac{{{\pi ^2}}}{4}$$
Work Step by Step
$$\eqalign{
& {\text{Calculate the volume using the disk method about the y - Axis}} \cr
& V = \int_c^d {\pi f{{\left( y \right)}^2}} dy \cr
& {\text{Let }}f\left( y \right) = \sin y,\,\,\,{\text{on the interval }}\left[ {0,\frac{\pi }{2}} \right] \cr
& V = \int_0^{\pi /2} {\pi {{\left( {\sin y} \right)}^2}} dy \cr
& V = \pi \int_0^{\pi /2} {{{\sin }^2}y} dy \cr
& V = \pi \int_0^{\pi /2} {\frac{{1 - \cos 2y}}{2}} dy \cr
& V = \frac{\pi }{2}\int_0^{\pi /2} {\left( {1 - \cos 2y} \right)} dy \cr
& {\text{Integrate}} \cr
& V = \frac{\pi }{2}\left[ {y - \frac{1}{2}\sin 2y} \right]_0^{\pi /2} \cr
& V = \frac{\pi }{2}\left[ {\frac{\pi }{2} - \frac{1}{2}\sin 2\left( {\frac{\pi }{2}} \right)} \right] - \frac{\pi }{2}\left[ {0 - \frac{1}{2}\sin 2\left( 0 \right)} \right] \cr
& {\text{Simplifying}} \cr
& V = \frac{\pi }{2}\left[ {\frac{\pi }{2} - 0} \right] - \frac{\pi }{2}\left[ {0 - 0} \right] \cr
& V = \frac{{{\pi ^2}}}{4} \cr} $$
