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} 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; then S is defined to be an upper Riemann sum. Similarly, if v_i is the infimum of f over x_i, then S is a lower Riemann sum.

Riemann Sum

See also