Here is the exact question: https://imgur.com/a/cBQC8su!
My particular question regards the range of $\phi$; $\phi$ certainly lives between $0$ $\le$ $\phi$ $\le$ $\frac {\pi}{2}$. $\rho = 1$ intersects with $ \rho = 2cos\phi$ at $\frac {\pi}{3}$. I thought that the range of $\phi$ would be $0$ $\le$ $\phi$ $\le$ $\frac {\pi}{2}$, where $0$ $\le$ $\phi$ $\le$ $\frac {\pi}{3}$ to $\frac {\pi}{3}$ $\le$ $\phi$ $\le$ $\frac {\pi}{2}$. Any helpful tips? The book lists the answer as $\frac {5\pi}{12}$.
The intersection $S$ of the two unit balls is a lens shaped object whose volume can easily be computed by the "washer method". One obtains $${\rm vol}(S)=2\cdot\int_{1/2}^1\pi(1-r^2)\>dr={5\pi\over12}\ .$$