We approximate the integral of a smooth function on [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 con-vergence is as fast as n3½, whereas the limiting distribution is
Gaussian then.