Riemann Sum

The Riemann sum of f over I with partition P is defined as

where x_i−1 ≤ x*_i ≤ x_i. The choice of x*_i in this interval is arbitrary. If x*_i = x_i−1 for all i, then S is called a left Riemann sum. If x*_i = x_i, then S is called a right Riemann sum. If x*_i = 1⁄₂(x_i+x_i−1), then S is called a middle Riemann sum. The average of the left and right Riemann sum is the trapezoidal sum.

If it is given that

where v_i is the supremum of f over, then S is defined to be an upper Riemann sum. Similarly, if v_i is the infimum of f over, then S is a lower Riemann sum.

Any Riemann sum on a given partition (that is, for any choice of x*_i between x_i−1 and x_i) is contained between the lower and the upper Riemann sums. A function is defined to be Riemann integrable if the lower and upper Riemann sums get ever closer as the partition gets finer and finer. This fact can also be used for numerical integration.