I am interested in numericaly integrating a function over the surface of a circle (x^2+y^2<1). As a constraint, the function is not analytic, and evaluating it is relatively expensive in terms of time. I don't know how accurately the integration will need to be done - this is one of the points of interest. Fortunately it has no singularities and should be relatively smooth (and is always positive, if this helps). I have got as far as the 22 point scheme in Abromowitz and Stegun (in 25.4.61), but I don't know how to extend this formultation.
Since a circle is a simple shape, I suspect that better weighting schemes are available. I would appreciate any suggestions about where to go for higher-order methods.