# Riemann integral

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## 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/