Having trouble calculating bounds and how to do it generally for other terms.
I've been asked to find the volume of the solid bounded by the sphere
$x^2 + y^2 + z^2 = 4$
and the cones
$z = \sqrt{x^2 + y^2}$
$z = -\sqrt{3x^2 + 3y^2}$
I assume that $0 <= \rho <= 2$ and $0 <= \theta <= 2\pi$, but I don't know how to find $\phi$ for either cones.
Using cylindrical coordinates, we have the equations $$\begin{align} z^2&=4-r^2,\\ z&=r\\ z&=-\sqrt 3 r \end{align}.$$
The first equation gives $4-r^2\ge 0,$ or $0\le r\le 2.$ Then the last two give $-\sqrt 3 r\le z\le r.$ Finally, we have $0\le \phi\le 2π,$ since all the equations are independent of $\phi.$
So you want to calculate the $$\int_0^{2π}\int_0^2\int_{-\sqrt 3 r}^r{r\mathrm dz \mathrm dr\mathrm d\phi.}$$