Problem
I am trying to compute the volume of the region bounded above by the hemisphere $z=(a^2-x^2-y^2)^{0.5}$ and below by the cone $z=(x^2+y^2)^{0.5}$
I know that the question is solvable using cylindrical and spherical coordinates and so I was trying both when I ran into a problem whilst attempting to solve using cylindrical coordinates.
The problem that I have is that I don’t seem to be identifying the correct limits for the problem.
The correct solution is as follows:
However, to me, this looks to only cover half of the shaded region. It feels natural to me to multiply this integral by 2 so that we consider the values of r between $r=\frac{-a}{\sqrt{2}}$ and $r=0$ which don’t seem to be considered in the limits of the solution integral.
I was wondering if anyone could explain where the gap in my intuition is here? To me it feels as though this integral neglects the left half of the shaded region in the graph above.
Thanks to @IvanKaznacheyeu for your comment helping me to identify my mistake.
Despite the fact that theta imposes no restrictions on the region, we cannot reduce this to a 2 dimensional integral as $\theta$ causes a rotation about the $z$ axis.
Therefore, the rotation for $\theta$ betweeen $[0,2\pi]$ means that the left hand side of the region is covered by the integral in the image and there is no need for us to double the value of V calculated there as this will lead to an incorrect answer (doubling the actual volume).