Consider the sphere of unit radius centered at $(0,0,1)$,and the cone of equation $z^2=x^2+y^2.$ Find the volume above the cone and inside the sphere.
The equation of the sphere is $$1 \leq x^2+y^2+(z-1)^2=x^2+y^2+z^2-2z+1$$
Using spherical coordinates this gives $$1 \leq \rho ^2\sin^2\varphi \cos^2\theta + \rho^2\sin^2\varphi sin^2\theta + \rho^2 \cos^2\varphi -2\rho \sin\varphi \sin\theta +1 $$
The solution says I should come to $\rho \leq 2\cos\varphi$. I don't understand where they have got this from though, could someone explain?