double and triple integrals boundary as the zero-level contour of the level-set function for simple geometries one can easily construct by handwhile in more complicated industrial applications one must resort to mathematical models for constructing let "be the area of domain we can estimate the integral by this monte carlo integration method embed the geometry in rectangular area draw large number of random points xyin count the fraction of points that are inside approximate "/= rby qi set " qa evaluate the mean of fnat the points inside estimate the integral as "/fn note that ris trivial to compute since is rectanglewhile "is unknown howeverif we assume that the fraction of roccupied by "is the same as the fraction of random points inside "we get simple estimate for "to get an idea of the methodconsider circular domain embedded in rectangle as shown below collection of random points is illustrated by black dots implementation python function implementing xy/dxdy can be written like thisimport numpy as np def montecarlo_double(fgx )""monte carlo integration of over domain >= embedded in rectangle [ , ] [ , ^ is the number of random points ""