I think you are all looking at the wrong angle. $\phi$ measures the angle from the $z$ axis to the surface. It is $\theta$ that goes from $0$ to $2\pi$ (the usual polar coordinate angle). You can tell that from the equation $r = g(\theta,z)$. For $\phi$ you have to go between the asymptotes. I would try $\phi \in (\frac \pi 4,\frac {3\pi} 4)$.