Integral of a function inside non-rectangular domain

[stevengj]

@wsshin

I want to integrate some function f(x,y) inside a non-rectangular domain, say a disk defined by x² + y² ≤ 1.

Have you considered cylindrical coordinates?

In general, you want to do some change of variables to map the domain to a hypercube (or a collection thereof). See also 2D integration over non-rectangular domain using cubature - #7 by stevengj

@wsshin

I actually tried to define g(x,y) = f(x,y) * (x^2+y^2 ≤1)

The problem with integrating this is that now you have a discontinuous integrand. Quadrature schemes are generally designed for smooth integrands. (If you have a singularity, you want to build it into the quadrature scheme if possible.)

@wsshin

However, when I tried this with HCubature package, I was not able to get a solution for several minutes

Realize that hcubature defaults to \sqrt{\varepsilon} tolerance — so it is trying to integrate your discontinuous integrand to 8 significant digits by refining the domain, and it probably won’t converge until it has > 10^8 subdomains; it’s not surprising that it takes a ridiculously long time.