Answer
$\displaystyle{V=2\sqrt2\pi-\frac{5\pi}{2}}$
Work Step by Step
$\displaystyle{A(x)=\pi(1-\sin x)^2-\pi(1-\cos x)^2}\\
\displaystyle{A(x)=\pi\left(2\cos x-2\sin x-\cos2x\right)}$
$\displaystyle{V=\int_0^{\frac{\pi}{4}}A(x)\ dx}\\
\displaystyle{V=\int_0^{\frac{\pi}{4}}\pi\left(2\cos x-2\sin x-\cos2x\right)\ dx}\\
\displaystyle{V=\pi\int_0^{\frac{\pi}{4}}2\cos x-2\sin x-\cos2x\ dx}\\
\displaystyle{V=\pi\left[2\sin x+2\cos x-\frac{1}{2}\sin2x\right]_0^{\frac{\pi}{4}}}\\
\displaystyle{V=\pi\left[2\sqrt2-\frac{5}{2}\right]}\\
\displaystyle{V=2\sqrt2\pi-\frac{5\pi}{2}}$
