How to describe the region inside a sphere and below a cone in cylindrical and spherical coordinates?

Question

If E is the region of space located inside the sphere $x^2 + y^2 + z^2 = 4$ and below the cone $z = \sqrt{3x^2 + 3y^2}$

How may I describe E in cylindrical and spherical coordinates? And how may I use E to evaluate:

\begin{equation*} \int\int\int_{E} z^2 dV \end{equation*}

I'm stuck, so any tip will be helpful

Thanks in advance!

Answer 1

In spherical coordinates, E can be described by $0\le\theta\le 2\pi, \;\;0\le\rho\le2, \;\;\frac{\pi}{6}\le\phi\le\pi$.

In cylindrical coordinates, I believe E can be described by $0\le\theta\le2\pi$ and

for $0\le r\le1, \;\;-\sqrt{4-r^2}\le z\le\sqrt{3}r$

for $1\le r\le2, \;\;-\sqrt{4-r^2}\le z\le\sqrt{4-r^2}$.