Riemann sum (original) (raw)

where cj∈[xj-1,xj] is chosen arbitrary.

If cj=xj-1 for all j, then S is called a left Riemann sum.

If cj=xj for all j, then S is called a Riemann sum.

Equivalently, the Riemann sum can be defined as

where bj∈{f⁢(x):x∈[xj-1,xj]} is chosen arbitrarily.

If bj=supx∈[xj-1,xj]⁡f⁢(x), then S is called an upper Riemann sum.

If bj=infx∈[xj-1,xj]⁡f⁢(x), then S is called a lower Riemann sum.

For some examples of Riemann sums, see the entry examples of estimating a Riemann integral.