# Riemann integral

Let $I=[a,b]$ be an interval of $\mathbb{R}$ and let $f:I\to \mathbb{R}$ be a bounded function. For any finite set^{} of points $\{{x}_{0},{x}_{1},{x}_{2},\mathrm{\dots},{x}_{n}\}$ such that $$, there is a corresponding partition^{} $P=\{[{x}_{0},{x}_{1}),[{x}_{1},{x}_{2}),\mathrm{\dots},[{x}_{n-1},{x}_{n}]\}$ of $I$.

Let $C(\u03f5)$ be the set of all partitions of $I$ with $$. Then let ${S}^{*}(\u03f5)$ be the infimum^{} of the set of upper Riemann sums with each partition in $C(\u03f5)$, and let ${S}_{*}(\u03f5)$ be the supremum of the set of lower Riemann sums with each partition in $C(\u03f5)$. If $$, then $C({\u03f5}_{1})\subset C({\u03f5}_{2})$, so ${S}^{*}(\u03f5)$ is decreasing (http://planetmath.org/IncreasingdecreasingmonotoneFunction) and ${S}_{*}(\u03f5)$ is increasing (http://planetmath.org/IncreasingdecreasingmonotoneFunction). Moreover, $|{S}^{*}(\u03f5)|$ and $|{S}_{*}(\u03f5)|$ are bounded by $(b-a){sup}_{x}|f(x)|$. Therefore, the limits ${S}^{*}={lim}_{\u03f5\to 0}{S}^{*}(\u03f5)$ and ${S}_{*}={lim}_{\u03f5\to 0}{S}_{*}(\u03f5)$ exist and are finite. If ${S}^{*}={S}_{*}$, then $f$ is Riemann-integrable over $I$, and the Riemann integral of $f$ over $I$ is defined by

$${\int}_{a}^{b}f(x)\mathit{d}x={S}^{*}={S}_{*}.$$ |

Title | Riemann integral |
---|---|

Canonical name | RiemannIntegral |

Date of creation | 2013-03-22 11:49:24 |

Last modified on | 2013-03-22 11:49:24 |

Owner | bbukh (348) |

Last modified by | bbukh (348) |

Numerical id | 14 |

Author | bbukh (348) |

Entry type | Definition |

Classification | msc 28-00 |

Classification | msc 26A42 |

Related topic | RiemannSum |

Related topic | Integral2 |

Defines | Riemann integrable |