Going Further

There are many other types of integration methods that are useful.

Gaussian Quadrature

A very powerful technique is Gaussian quadrature of which Gauss-Legendre quadrature is one of the most popular techniques. By using \(N\) points, Gauss-Legendre can exactly integrate a polynomial of order \(2N - 1\)!

Monte Carlo Integration

For multi-dimensional integration, we can apply our composite integration rules in succession over each dimension. But when the dimension is really high (\(N > 3\)), this becomes computationally expensive and randomly sampling the integrand becomes better—this is Monte Carlo integration.