Multi-dimensional Integration

In principle, one can extend the one-dimensional integration methods to more than one dimensions. This is indeed trivial if the integration boundary in multi-dimensional integrations is simple ( say a multi-dimensional cube or sphere or some such object ). If the boundary is complicated, one needs to incorporate that in the integration scheme. That may be difficult but can still be programmed. The real difficulty comes when the dimensionality is large. One then needs large number of integration points to get a reasonable accuracy for the integral. Typically, if one needs 10 points for 1% accuracy in one dimension, for N-dimensional integration, one would expect $10^N$ integration points to achieve similar accuracy. This runs into millions of points for more than six dimensions. At this stage, it may be useful to use Monte Carlo methods to estimate the integral. Generally, for Monte Carlo methods with N points, the accuracy goes as $\frac{1}{\sqrt{N}}$. So with a million points in N dimensions, we would expect an accuracy of 1% ( at least ). We shall discuss Monte Carlo method when we discuss random numbers.