I trying to find the volume of the region in $R^3$ satisfying
$ 0 < \sqrt{x^2+y^2} \leq x^2+y^2 \leq z\sqrt{x^2+y^2} \leq 2 \sqrt{x^2+y^2} + y $
Not even sure whether i should use cylindrical coordinate or spherical coordinate especially beacuse of the last inequality. I'm having a tough time, is there any help?
Use cylindrical coordinates, I would say. The conditions become $$ 0<r\le r^2 \le zr \le 2r +r\cos\theta, $$or$$ 1\le r \le z \le 2 +\cos\theta$$ Ignoring the expression with $z$ for the moment, we see that the solid in question lies above the region in the $xy-$plane outside the unit circle and inside the limaçon $ r=2+\cos\theta,$ so that the region to integrate over will be determined by the inequalities $$ 0\le\theta\le2\pi,\ 1\le r \le 2+\cos\theta$$
You take it from here.