Find the volume of the intersection of the two cylinders
$$x^2 + z^2 =9 \
and\
y^2 + z^2 =9
$$

Question

anonymous · Answer

Here is how this volume looks like

anonymous · Answer

I wish I weren't so rusty with this stuff, lol. Do you happen to have an answer? I worked it out, but being rusty with it, it would be great to get confirmation.

anonymous · Answer

Due to symmetry, the total volume can be expressed as a multiple of the volume of a half-octant (I don't think there's word to describe a sixteenth of the xyz space) of the intersection:

Then you have
$$\large V=16\int_0^{\pi/4}\int_0^{\pi/2}\int_0^{3\sqrt2} \rho^2\sin\phi~d\rho~d\phi~d\theta$$
I *think*. I haven't worked with spherical coordinates in a while - I'm pretty confident in the ranges for $\theta$ and $\phi$, but not so much for $\rho$. Something tells me it might be $3\le\rho\le3\sqrt2$...

anonymous · Answer

Oops, meant to attach a picture. I tried to use as vibrant colors as I could find.