Because of the obvious cylindrical symmetry, I would recommend changing to polar coordinates in the yz-plane. Then , and the cylinder, becomes r= 2 and x= 2y gives height .

Integrate with r ranging from 0 to 2 and ranging from 0 to .

(Normally, of course, we use polar coordinates in an xy-plane. Essentially what I did was swap x and z so the problem would be "find the area of the solid bounded by in the first quadrant from z= 0 to z= 2y. Changing the coordinate system doesn't change the solid so the two solids have the same volume)