Answer
$$V = \frac{\pi }{2}$$
Work Step by Step
$$\eqalign{
& {\text{Let }}f\left( x \right) = \cos x{\text{ and }}g\left( x \right) = \sin x \cr
& {\text{The volume is given by}} \cr
& V = \pi \int_a^b {\left( {f{{\left( x \right)}^2} - g{{\left( x \right)}^2}} \right)dx} \cr
& V = \pi \int_0^{\pi /4} {\left( {{{\cos }^2}x - {{\sin }^2}x} \right)dx} \cr
& {\text{Use the identity }}\cos 2x = {\cos ^2}x - {\sin ^2}x \cr
& V = \pi \int_0^{\pi /4} {\cos 2x} dx \cr
& {\text{Integrating}} \cr
& V = \pi \left[ {\frac{1}{2}\sin 2x} \right]_0^{\pi /4} \cr
& V = \frac{\pi }{2}\left[ {\sin 2\left( {\frac{\pi }{4}} \right) - \sin 2\left( 0 \right)} \right] \cr
& V = \frac{\pi }{2} \cr} $$
