Find the volume of the solid formed by intersecting the sphere $x^ 2 +y ^2 +z ^2 = a^ 2$ with the cylinder $x ^2 +y ^2 = ax$. Find the volume of the solid formed by intersecting the sphere $x^2 +y^2 +z^2 = a^ 2$ with the cylinder $x^2 +y^2 = ax$.

Hint. By symmetry,
$$\mbox{Volume}=2\int_D \sqrt{a^2-(x^2+y^2)}\,dx\,dy$$
where $D=\{(x,y)\in\mathbb{R}\;:\; x^2 +y^2 \leq ax\}$.
Now it is convenient  to write the integral in polar coordinates.
Note that $x^2 +y^2\leq ax$ implies that $\rho^2 \leq a\rho \cos(\theta)$ and, for $a>0$, that $\theta\in[-\pi/2,\pi/2]$. 
Hence the integral becomes
$$\mbox{Volume}=2\int_{-\pi/2}^{\pi/2}\int_{\rho=0}^{a\cos(\theta)} \sqrt{a^2-\rho^2}\,\rho d\rho d \theta.$$
Can you take it from here?
