Answer
$$V = {\pi ^2} - 2\pi $$
Work Step by Step
$$\eqalign{
& y = \cos x,{\text{ }}y = 0,{\text{ }}x = 0,{\text{ }}x = \pi /2 \cr
& {\text{Calculate the Volume by cylindrical shells about the }}y{\text{ - axis}} \cr
& V = 2\pi \int_a^b {xf\left( x \right)} dx,{\text{ then}} \cr
& {\text{The Volume of the solid is given by}} \cr
& V = 2\pi \int_0^{\pi /2} {x\cos x} dx \cr
& {\text{Integrating using the formula }}\int {u\cos u} du = \cos u + u\sin u + C \cr
& V = 2\pi \left[ {\cos x + x\sin x} \right]_0^{\pi /2} \cr
& V = 2\pi \left[ {\cos \left( {\frac{\pi }{2}} \right) + \frac{\pi }{2}\sin \left( {\frac{\pi }{2}} \right)} \right] - 2\pi \left[ {\cos \left( 0 \right) + \frac{\pi }{2}\sin \left( 0 \right)} \right] \cr
& {\text{Simplifying}} \cr
& V = 2\pi \left[ {\frac{\pi }{2}} \right] - 2\pi \left[ {1 + 0} \right] \cr
& V = {\pi ^2} - 2\pi \cr} $$
