# Is it possible to solve a Steinmetz solid with cylindrical coordinates?

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.