We can't help you because the problem is missing an important piece of information (the limits for r). Please post the whole problem.
Could you switch the order of integration on this? If are constant wrt z and r I don't see the difference in integrating wrt z first or wrt r first. If that is the case you could just integrate wrt z and then let the OP use the limits they have wrt r. (I guess either way there would need to be an extra piece of information, whether the above vars are constant wrt r and z or the limits on r).