Integrate with respect to x and you only have to calculate one integral (for cylindrical shells).

Alternatively, your method is fine if you either now calculate and remove a central cylindar (radius one around the y axis) or go back and adust the integrands so that your Reimann strips aren't coming right down to the wire (the y axis) but only to x = 1, and so that they rotate to form washers instead of disks. (Look again at your sketch.)