How to determine $\varphi$ in spherical coordinates

Question

Assume that I would like to integrate some continuous a.e. function $f(x,y,z)$ over the following set: $ a^2_1 \le x^2 + y^2 +z^2 \le a^2_2$, and $z\ge c^2(x^2+y^2)^{1/2}$. So, in a case I would like to convert to spherical coordinates, then clearly $a_1^2 \le x^2 + y^2 \le a_2^2$, hence $a_1 \le r \le a_2$, and because the projection on the $x\times y$ plane is two circles of radius $a_1$ and $a_2$, then $\theta \in [0, 2\pi]$. But how to determine $\varphi$ in this case?

Answer 1

The inequality $z \geq c^2(x^2+y^2)^{1/2}$ is equivalent to $\cot(\varphi) \geq 1$. Since $\cot$ is decreasing this is equivalent to $\operatorname{arccot}(\varphi) \leq \pi/4$.