Riemann sum

In mathematics, a Riemann sum is a method for approximating the values of integrals.

Let it be supposed there is a function f: D → R where D, R ⊆ R and that there is a closed interval I = [a,b] such that I ⊆ D. If we have a finite set of points {x₀, x₁, x₂, ... x_n} such that a = x₀ < x₁ < x₂ ... < x_n = b, then this set creates a partition P = {[x₀, x₁), [x₁, x₂), ... [x_n-1, x_n]} of I.

If $P$ is a partition with $n \in \mathbb{N}$ elements of $I$ , then the Riemann sum of $f$ over $I$ with the partition $P$ is defined as

$S = \sum_{i=1}^{n} f(y_i)(x_{i}-x_{i-1})$

where x_i-1 ≤ y_i ≤ x_i. The choice of y_i is arbitrary. If y_i = x_i-1 for all i, then S is called a left Riemann sum. If y_i = x_i, then _S is called a right Riemann sum.

Suppose we have

$S = \sum_{i=1}^{n} v_i(x_{i}-x_{i-1})$

where v_i is the supremum of f over [x_i-1, x_i]; then S is defined to be an upper Riemann sum. Similarly, if v_i is the infimum of f over [x_i-1, x_i], then S is a lower Riemann sum.

Riemann sum

See also

See also: Riemann sum, Bernhard Riemann, Closed interval, Function (mathematics), Infimum, Integral, Mathematics, Partition of a set, Real number, Riemann-Stieltjes integral