I have an algorithm for solving a certain problem that requires that I compute two-dimensional integrals as a subroutine, and I'd like to make some kind of statement about its running time. Suppose $S$ is a simply connected region in the plane whose boundary is composed of $n$ line segments and $m$ arcs of some type (say, arcs of a circle, ellipse, or parabola). I'd like to numerically approximate the integral of some function $f(x)$ over $S$. What's the "computational complexity", in big-O notation, of computing this within a given error $\epsilon$? Clearly this depends on lots of factors, such as the curvature of the arcs and the derivatives of the function $f(x)$ but I haven't had much success in pinning down anything concrete. By way of comparison, the wikipedia page for Simpson's rule

http://en.wikipedia.org/wiki/Simpson%27s_rule#Error

says that the error committed by the composite Simpson's rule is bounded (in absolute value) by

$\frac{h^4}{180} (b-a) M$

where $h$ is the step size and $M = \max_{\xi\in[a,b]}|f^{(4)}(\xi)|$, and therefore the complexity of computing such an integral within precision $\epsilon$ is $M^{1/4}(b-a)^{5/4}\epsilon^{-1/4}$. Can I make any similar statement for a two-dimensional integral over my region $S$? What if I were able to triangulate $S$ into $N$ small pieces?