Find the volume of the region that lies inside the cylinder $x^2 + y^2 = 1$ and $x^2 + y^2 + z^2 = 4.$
I've attempted to break this down into 2 sections: a pure cylinder, which continues until it hits the edges of the sphere, and then 2 "caps." I believe the overall shape looks like a pill. The integrals I've arrived at look like this:
For the cylinder:
$\int_{0}^{2\pi}\int_{0}^{1}\int_{-\sqrt3}^{\sqrt3}rdzdrd\theta$
And for the caps:
$2\int_{0}^{2\pi}\int_{0}^{\frac{\pi}{6}}\int_{0}^{2}\rho^2 sin\phi d\rho d\phi d\theta$
For the cylinder, I simply plugged $x^2 + y^2 = 1$ into $x^2 + y^2 + z^2 = 4$ to arrive at $r= -\sqrt3, \sqrt3$. For the sphere, I got $\phi=\frac{\pi}{6}$ based on the fact that the arm is $\sqrt3$ in the z direction and 1 in the radial direction (based on the cylinder). However, the answer I get with these integrals is wrong. Where am I screwing this up?