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In mathematics, a Riemann sum is a method for approximating the values of integrals. It may also be used to define the integration operation. The sums are named after Bernhard Riemann. There are four different kinds of Riemann sums, called left, right, middle and trapezoidal sum. Any Riemann sum on a given partition 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. More on Riemann sums can be found in the free eBook Elementary Calculus by H.J. Keisler.