Riemann Sum

Riemann Sum
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Let a closed interval [a,b] be partitioned by points a<x_1<x_2<...<x_(n-1)<b, where the lengths of the resulting intervals between the points are denoted Deltax_1, Deltax_2, ..., Deltax_n. Let x_k^* be an arbitrary point in the kth subinterval. Then the quantity


is called a Riemann sum for a given function f(x) and partition, and the value maxDeltax_k is called the mesh size of the partition.

If the limit of the Riemann sums exists as maxDeltax_k->0, this limit is known as the Riemann integral of f(x) over the interval [a,b]. The shaded areas in the above plots show the lower and upper sums for a constant mesh size.

See also

Integral, Lower Sum, Mesh Size, Riemann Integral, Upper Sum Explore this topic in the MathWorld classroom

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Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, pp. 324-327, 1999.

Referenced on Wolfram|Alpha

Riemann Sum

Cite this as:

Weisstein, Eric W. "Riemann Sum." From MathWorld--A Wolfram Web Resource.

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