I am not sure if I am doing the right thing here in setting up the integrals using cylindrical coordinates. I need to set up the integral using cylindrical coordinates to solve for the volume such that the triple integral is $\iiint xyz \,dV$ and S is the region bounded by the cylinders $x^2+y^2=25$, $x^2+z^2=25$, and the first octant.
My attempt here is that I converted the cylinders to its cylindrical coordinates; hence, it become
$$ x^2+y^2=25 \implies r =5$$ $$ x^2+z^2 = 25 \implies r^2\cos^2(\theta) + z^2 = 25$$
If I consider the intersection $ y = \pm z$, then are the bounds for z here will be $0 \leq z \leq r \sin(\theta)$ and the bounds for r here will be $0 \leq r \leq 5$?