# Riemann integral

###### This wiki is incomplete.

## Definition

The Riemann integral of $f$ equals $S$ if the following condition holds:

For all $\epsilon > 0$, there exists a $\delta$ such that for any tagged partition $x_0, \ldots , x_n$ and $t_0, \ldots , t_{n-1}$ whose mesh is less than $\delta$, we have

$| \sum_{i=0}^{n-1} f(t_i) ( x_{i+1} - x_i) - S | < \epsilon$

## Properties

(Proof from first principle)

## Reasons of failure

Evaluate the Riemann integral

$\int_0^1 \frac{1}{ \sqrt{x} } \, dx .$

**Note**: The Riemann integral of $f$ equals $S$ if the following condition holds:

$\forall \epsilon > 0$, $\exists \delta$ such that for any tagged partition $x_0, \ldots , x_n$ and $t_0, \ldots , t_{n-1}$ whose mesh is less than $\delta$, we have

$\left| \sum_{i=0}^{n-1} f(t_i) ( x_{i+1} - x_i) - S \right| < \epsilon.$

Inspiration.

## Riemann-Lebesgue criterion

The function $f: [ a , b ] \rightarrow \mathbb{R}$ is Riemann integrable if and only if (it is bounded and the set of points at which it is discontinuous has Lebesgue measure zero).

**Cite as:**Riemann integral.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/riemann-integral/