Riemann sum
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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, ... xn} such that a = x0 < x1 < x2 ... < xn = b, then this set creates a partition P = {[x0, x1), [x1, x2), ... [xn-1, xn]} 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 xi-1 ≤ yi ≤ xi. The choice of yi is arbitrary. If yi = xi-1 for all i, then S is called a left Riemann sum. If yi = xi, 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 vi is the supremum of f over [xi-1, xi]; then S is defined to be an upper Riemann sum. Similarly, if vi is the infimum of f over [xi-1, xi], then S is a lower Riemann sum.