Monte Carlo Integration

In order to integrate a function over a complicated Domain $D$, Monte Carlo integration picks random points over some simple Domain $D'$ which is a superset of $D$, checks whether each point is within $D$, and estimates the Area of $D$ (Volume, $n$-D Content, etc.) as the Area of $D'$ multiplied by the fraction of points falling within $D'$.

An estimate of the uncertainty produced by this technique is given by

$$\int f\,dV \approx V\left\langle{f}\right\rangle{}\pm \sqrt{... ...{f^2}\right\rangle{}-\left\langle{f}\right\rangle{}^2\over N}.$$

See also Monte Carlo Method

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Simple Monte Carlo Integration'' and ``Adaptive and Recursive Monte Carlo Methods.'' §7.6 and 7.8 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 295-299 and 306-319, 1992.