I've been looking for a way to calculate the volume of a two cylinder intersection using cylindrical coordinates, but I can't find a way to make it work. I've been able to calculate the volume with cartesian coordinates and with a formula and I know the correct result ($\frac{1024}{3}u^3$) but I can't get to the same result with cylindrical coordinates.
I've been trying to use this integration limits:
$0<\theta<2\pi$
$0<r<4$
$-\sqrt{16-r^2cos^2\theta} < z < \sqrt{16-r^2cos^2\theta}$
For the cylinders: $x^2+y^2=16$ and $x^2+z^2=16$
I think that I have set the integration limits wrong but I don't know which ones to use.
$$ x^2 + y^2 = 16\\ x^2 + z^2 = 16 $$ That suggests a strong symmetry about $x$, so you should change coordinates, and write $$ z^2 + y^2 = 16\\ z^2 + x^2 = 16 $$ which is essentially the same solid. Now each $z =$constant slice is a square.
At that point, instead of computing the entire integral, you should integrate from $\theta = -\pi/4$ to $\theta = \pi/4$, which will give you $1/4$ of the total, but has the advantage that on that sector, $x$ is a constant.
The short answer, though, is "draw a picture and think about it for a while, and you'd have come to the same conclusion that I did here."