In order to integrate a function over a complicated domain, Monte Carlo integration picks random
points over some simple domain which is a superset of , checks whether each point is within , and estimates the area of (volume, -dimensional content, etc.) as
the area of multiplied by the fraction of points falling within . Monte Carlo integration is implemented
in the Wolfram Language as NIntegrate[f,
..., Method -> MonteCarlo].
Picking
randomly distributed points , , ..., in a multidimensional volume to determine the integral of a function in this volume gives a result
Hammersley, J. M. "Monte Carlo Methods for Solving Multivariable Problems." Ann. New York Acad. Sci.86, 844-874,
1960.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.Ueberhuber,
C. W. "Monte Carlo Techniques." §12.4.4 in Numerical
Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag,
pp. 124-125 and 132-138, 1997.Weinzierl, S. "Introduction
to Monte Carlo Methods." 23 Jun 2000. http://arxiv.org/abs/hep-ph/0006269.
, Monte Carlo integration picks random
points over some simple domain 
which is a superset of 
, checks whether each point is within 
, and estimates the area of 
(volume, 
-dimensional content, etc.) as
the area of 
multiplied by the fraction of points falling within 
. Monte Carlo integration is implemented
in the Wolfram Language as NIntegrate[f,
..., Method -> MonteCarlo].
randomly distributed points 
, 
, ..., 
in a multidimensional volume 
to determine the integral of a function 
in this volume gives a result
