The **Riemann sum** of *f* over *I* with partition *P* is defined as

where *x*_{i−1} ≤ *x**_{i} *≤* x_{i}*. The choice of* x*_{i} in this interval is arbitrary. If *x**_{i} = *x*_{i−1} for all *i*, then *S* is called a **left Riemann sum**. If *x**_{i} = *x*_{i}, then *S* is called a **right Riemann sum**. If *x**_{i} = 1⁄_{2}(*x*_{i}+*x*_{i−1}), then *S* is called a **middle Riemann sum**. The average of the left and right Riemann sum is the **trapezoidal sum**.

If it is given that

where *v*_{i} is the supremum of *f* over, then *S* is defined to be an **upper Riemann sum**. Similarly, if *v*_{i} is the infimum of *f* over, then *S* is a **lower Riemann sum**.

Any Riemann sum on a given partition (that is, for any choice of *x**_{i} between *x*_{i−1} and *x*_{i}) 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.

Read more about Riemann Sum: Methods, Example, Animations

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