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.