Answer
$$V = \pi $$
Work Step by Step
$$\eqalign{
& y = \cos \frac{x}{2},{\text{ }}y = \sin \frac{x}{2},{\text{ }}x = 0,{\text{ }}x = \frac{\pi }{2} \cr
& {\text{Using the washer method}} \cr
& V = \pi \int_a^b {\left( {{{\left[ {R\left( x \right)} \right]}^2} - \left[ {r\left( x \right)} \right]} \right)} dx \cr
& {\text{From the graph }}R\left( x \right) = \cos \frac{x}{2}{\text{ and }}r\left( x \right) = \sin \frac{x}{2} \cr
& V = \pi \int_0^{\pi /2} {\left( {{{\cos }^2}\frac{x}{2} - {{\sin }^2}\frac{x}{2}} \right)} dx \cr
& {\text{Use the trigonometric identity co}}{{\text{s}}^2}x - {\sin ^2}x = \cos 2x \cr
& V = \pi \int_0^{\pi /2} {\cos 2\left( {\frac{x}{2}} \right)} dx \cr
& V = \pi \int_0^{\pi /2} {\cos x} dx \cr
& {\text{Integrate}} \cr
& V = \pi \left[ {\sin x} \right]_0^{\pi /2} \cr
& V = \pi \left[ {\sin \left( {\frac{\pi }{2}} \right) - \sin \left( 0 \right)} \right] \cr
& V = \pi \left( {1 - 0} \right) \cr
& V = \pi \cr} $$
