(Couldn't upload the picture showing the integral as I'm a new user.)
Yes. Those integrals (I'll assume they're area integrals over a region in 2D space) can be calculated using an appropriate quadrature rule.
You can also use Green's theorem to convert them into contour integrals and use Gaussian quadrature to integrate along the path.
Thanks duffymo! I am calculating Huygens-Fresnel diffraction integrals: plane and other wave diffraction through circular (2D) apertures in polar coordinates.
As far as the programming goes: Currently a lot of my code is in Mathematica. I am considering changing to one of: scipy, java + flanagan math library, java + apache commons math library, gnu scientific library, or octave.
My first candidate for evaluation is scipy, but if it cannot handle complex-valued integrands, then I have to change my plans for the weekend...