Riemann sum

In mathematics, let it be supposed there is a function <math>f: D \rightarrow R</math> where <math>D, R \subseteq \mathbb{R}</math> and that there is a closed interval <math>I = [a,b]</math> such that <math>I \subseteq D</math>. If we have a finite set of points <math>\{x_0, x_1, x_2, \dots x_n\}</math> such that <math>a = x_0 < x_1 < x_2 \dots < x_n = b</math>, then this set creates a partition <math>P = \{[x_0, x_1), [x_1, x_2), \dots [x_n-1, x_n]\}</math> of <math>I</math>.

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 <math>x_{i-1} \leq y_i \leq x_i</math>. The choice of <math>y_i</math> is arbitrary. If <math>y_i = x_{i-1}</math> for all <math>i</math>, then <math>S</math> is called a left Riemann sum. If <math>y_i = x_i</math>, then <math>S</math> is called a right Riemann sum.

Suppose we have

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

where <math>b</math> is the supremum of <math>f</math> over <math>[x_{i-1}, x_{i}]</math>; then <math>S</math> is defined to be an upper Riemann sum. Similarly, if <math>b</math> is the infimum of <math>f</math> over <math>[x_{i-1}, x_{i}]</math>, then <math>S</math> is a lower Riemann sum.

See also:

• Bernhard Riemann
• Riemann integral
• Riemann-Stieltjes integral[?]