On the approximation of an integral by a sum of random variables

We approximate the integral of a smooth function of [0, 1], where values are only known at n random points (i.e., a random sample from the uniform-(0, 1) distribution), and at 0 and 1. Our approximations are based on the trapezoidal rule and Simpson's rule (generalized to the non-equidistant case), respectively. In the first case, we obtain an n²-rate of convergence with a degenerate limiting distribution; in the second case, the rate of convergence is as fast as n^31/2, whereas the limiting distribution is Gaussian then.