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# Riemann Sum

 Graph the Riemann sum of as x goes from to using rectangles taking samples at the Maximum Minimum Left Right Midpoint Print estimated and actual areas? Rectangle Color Plot Color Light Gray Dark Gray Black White Red Orange Yellow Green Blue Purple Light Gray Dark Gray Black White Red Orange Yellow Green Blue Purple

Let a closed interval be partitioned by points , where the lengths of the resulting intervals between the points are denoted , , ..., . Let be an arbitrary point in the th subinterval. Then the quantity

is called a Riemann sum for a given function and partition, and the value is called the mesh size of the partition.

If the limit of the Riemann sums exists as , this limit is known as the Riemann integral of over the interval . 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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## References

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

Riemann Sum

## Cite this as:

Weisstein, Eric W. "Riemann Sum." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RiemannSum.html