Answer
$\frac{\pi}{2}$
Work Step by Step
The volume is given by
$$
\pi \int_{0}^{\pi / 2}\left(\sqrt{\cos x \sin x)^{2}} d x=\pi \int_{0}^{\pi / 2}(\cos x \sin x) d x\\=\frac{\pi}{2} \int_{0}^{\pi / 2} \sin 2 x d x=\left.\frac{\pi}{4}(-\cos 2 x)\right|_{0} ^{\pi / 2}=\frac{\pi}{2}\right.
$$
