Prove that a hyperbolic circle with hyperbolic radius in the hyperbolic upper half plane is

The problem suggests to place a coordinate system at the Euclidean of the circle and work out the double integral in polar coordinates. However, and I believe I must be messing up somewhere, if we let the Euclidean center of the circle be and have Euclidean radius , then we have in polar coordinates centered at the Euclidean center and , and so we have the hyperbolic area of the circle being:

However, this integral seems to be undefined because along the line somewhere, I get a , which is undefined at and .

Are my parameterizations wrong? Or are they correct and this integral does exist? Should I not deal with the Euclidean radius and approach it some other way? Thanks.