How do I convert this to spherical coordinates?

Question

The task is to set up the integral in spherical coordinates. The given integral is: $$\int_0^2\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}\int_{\sqrt{2x^2+2y^2}}^{\sqrt{8}} z^2-x^2-y^2 dzdydx $$ I'm new to spherical coordinates so I get confused as to what my bounds for $\phi$ would be? I'm getting $\phi=[0,\arctan(\frac{1}{\sqrt{2}})]$ Is that correct?

Answer 1

This figure showing the integration region should help:

$- \pi/2 \leq \theta \leq + \pi /2$

$0 \leq z \leq \sqrt{8}$

$0 \leq \phi \leq \tan^{-1} (1/\sqrt{2})$