Triple integrals(calculating volume)

Question

I want to check whether I am working this example correctly. I need to find volume of surface bounded with $y^2+z^2\le 2y,\sqrt{y^2+z^2}\le x, x\le 6-y^2-z^2$. First I graphed this in 3D. All of three given surfaces are located on x-axis. First one is inside of cylinder, second outside of upper cone, and third one paraboloid but only its inner part. So we have that the projection on YZ plane is the circle with radius 1, and center $(1,0)$. So i can use triple integrals to calculate the volume(and polar coordinates) so I get that $0\le r\le 2\cos\phi, -\frac{\pi}{2}\le\phi\le\frac{\pi}{2}, r\le x\le 6-r^2$. And what is left is to plug this in formula $V=\iiint dxdydz$. Is my working correct? Thank you.