Change of Variables & the Jacobian · Challenge problem

The solid torus $T$ is $\left(\sqrt{x^2 + y^2} - 3\right)^2 + z^2 \le 1$. Using the coordinates $x = (3 + s\cos\alpha)\cos\theta$, $y = (3 + s\cos\alpha)\sin\theta$, $z = s\sin\alpha$ with $0 \le s \le 1$, find the volume of $T$.