I'm wondering why my computation doesn't work and if someone could correct me on my misunderstanding.I got the correct answer via Spherical coordinates, but not by cylindrical.
Find the volume enclosed by the surfaces $x^2 + y^2 + z^2 = 2^2$ and $x^2 + y^2 + (z-2)^2 = 2^2$ and $\rho$ is the angle from positive $z$-axis.
This is my working:
In spherical coordinates,
$$V = \int_{0}^{2\pi} \int_{0}^{\frac{\pi}{3}} \int_{0}^{2} \rho ^2 \sin \phi d\rho d\theta d\phi + \int_{0}^{2\pi} \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \int_{0}^{4 \cos \phi} \rho ^2 \sin \phi d\rho d\theta d\phi = \frac{10\pi}{3}.$$
Now, my derivation in cylindrical coordinates gives me:
$$V = \int_{0}^{2\pi} \int_{0}^{2} \int_{2 - \sqrt{4-r^2}}^{\sqrt{4-r^2}} r dz dr d\theta = \frac{8\pi}{3}.$$
I've done this repeatedly, even with cartesian coordinates (which support my cylindrical answer, but I know the answer for spherical is correct).