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

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

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

<math>S = \sum_{i=1}^{n} v_i(x_{i}-x_{i-1})<math>

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.

See also

• Bernhard Riemann
• Riemann integral
• Riemann-Stieltjes integral