To start with, I expect that you have written the second bound wrongly, since that is the equation of a plane figure, not a solid...
Assuming the cylinder extends indefinitely up and down the z dimension, we have Viviani's Curve. Doing just the top half, we have z going from 0 (where it 'starts', on the (x,y) plane) up to (where it hits the hemisphere). And we have r going from 0 at the centre (z axis), up to a cos theta i.e. everywhere inside the cylinder. And theta is turning through the x-positive half of the space, i.e. from minus pi/2 to pi/2. So...
Just in case a picture helps to follow through from the inside out, we can start bottom left here, integrating r with respect to z...
... where (key in spoiler) ...
Spoiler:
Which leaves a couple of blanks to fill. Hope this helps.
_________________________________________
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!