Computing Integrals

A Crude Monte Carlo estimate of the integral

            ( 
        I = /  f(x)dx
            )A

is

  I ~ V(A)[f(x1)+... +f(xm)]/m

where V(A) is the size (measure) of the region A and x₁,...,x_m are uniformly distributed points in A, a subset of Rⁿ. If it is not possible to directly sample uniformly in A but A can be subscribed by a box H, sampling m uniform points in A can be obtained by sampling uniformly in H and accepting those m points that falls in A. By this V(A) can be estimated as
V(A) ~ V(H)·m/M
where M is the number of points sampled in H.

This technique to compute V(A) is called Hit-or-miss Monte Carlo and is very useful for computing measures of complicated regions.