## Calculus (3rd Edition)

$\pi^3-4 \pi$
The volume of a revolution can be calculated by using integration by parts as follows: $V=\pi \int_0^{\pi} (x \sqrt {\sin x})^2\\=\pi \int_0^{\pi} x^2 \sin x \ dx \\=\pi (-\pi^2 \cos \pi+0) +2 \pi \int_0^{\pi} x \cos x \ dx \\=\pi^3 +2 \pi \int_0^{\pi} x \cos x \ dx \\=\pi^3 +2 \pi (x |\sin x|_0^{\pi} -\int_0^{\pi} \sin x \ dx) \\=\pi^3 -2 \pi \int_0^{\pi} \sin x \ dx \\=\pi^3 +2 \pi |\cos x|_0^{\pi} \\=\pi^3-4 \pi$