By completing the square, that cylinder is which has axis (0, 1, z) which is slightly of center from the sphere. I think the first thing I would do is "translate" the origin to (0, 1, 0)- replace y by y'= y- 1 so that y= y'+ 1 and the equation of the cylinder is and the equation of the sphere is or .

Now the cylinder has its axis at (0, 0, z) and you can use cylindrical coordinates: z will run from the part of the sphere at the bottom, , to the part at the top, while r and range from 0 to 1 and 0 to . The volume is given by