[kens]

I was wondering the same thing, but near the end, the article discusses using statistical techniques to determine the standard error. In other words, you can easily get an idea of the accuracy of the result, which is harder with typical numerical integration techniques.

[ogogmad]

Numerical integration using interval arithmetic gets you the same thing but in a completely rigorous way.

[fph]

With many quadrature rules (e.g. trapezoidal rule, Simpson's rule) you have a very cheap error estimator obtained by comparing the results over n and 2n subdivision points.