This applet demonstrates elementary Monte Carlo integration of a two-dimensional function. The quantity integrated is the mean square distance from the origin for points within the enclosed region.

By clicking on the appropriate button, you can evaluate the integral via either the two-dimensional
rectangle rule or by simple Monte Carlo inetgration (no biased sampling).
In either case, the number of quadrature points *N*used to estimate the
integral can be varied by changing the value in the box (the largest value accepted is
10000).

The rectangle rule takes the quadrature points from a uniform grid, while the MC integration proceeds with points sampled at random uniformly over the integration region.

The result of the calculation is displayed, along with the error in the result After each integration, superposed on the figure are the (x,y)-values used to perform the quadrature

Repeated pressing of the buttons generates new samples, with different x-values taken to give new estimates of the integral.