Partitions for Optimal Approximations

The Riemann integral can be approximated using partitions and a rule for assigning weighted sums of the function at points determined by the partition. Approximation methods commonly used include endpoint rules, the midpoint rule, the trapezoid rule, Simpson’s rule, and other quadrature methods. The rate of approximation depends to a large degree on the rule being used and the smoothness of the function, but it also depends on the partition. We show that when one chooses an optimal partition, one gets a precise asymptotic rate of approximation and characteristic distribution of the points in the partition.