I want to find the volume of the solid bounded by the region $$E = \{(x,y,z) \, : \, x^2 + y^2 +z^2 - 2z \leq 0, \sqrt{x^2+y^2} \leq z\}$$ in spherical coordinates.
After setting up the region, my limits of integration are
$$0 \leq \rho \leq 2 \cos\phi$$ $$0 \leq \phi \leq \tfrac{\pi}{4}$$ $$0 \leq \theta \leq 2\pi$$
So evaluating the integral in spherical coordinates, I get
$$\int_{0}^{2\pi} \int_{0}^{\pi/4} \int_{0}^{2cos\phi} \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta = \pi$$
However, the solution has $\frac{\pi}{4}$ and the limits of $\theta$ are $0$ and $\frac{\pi}{2}$, and I don't understand why. I cannot figure out why $\theta$ has a maximum of $\frac{\pi}{2}$. I would appreciate it if someone would take a look at this problem for me. Thanks.