# Riemann sum

Let $I=[a,b]$ be a closed interval^{}, $f:I\to \mathbb{R}$ be bounded^{} on $I$, $n\in \mathbb{N}$, and $P=\{[{x}_{0},{x}_{1}),[{x}_{1},{x}_{2}),\mathrm{\dots}[{x}_{n-1},{x}_{n}]\}$ be a partition of $I$. The *Riemann sum ^{}* of $f$ over $I$ with respect to the partition $P$ is defined as

$$S=\sum _{j=1}^{n}f({c}_{j})({x}_{j}-{x}_{j-1})$$ |

where ${c}_{j}\in [{x}_{j-1},{x}_{j}]$ is chosen arbitrary.

If ${c}_{j}={x}_{j-1}$ for all $j$, then $S$ is called a *left Riemann sum*.

If ${c}_{j}={x}_{j}$ for all $j$, then $S$ is called a * Riemann sum*.

Equivalently, the Riemann sum can be defined as

$$S=\sum _{j=1}^{n}{b}_{j}({x}_{j}-{x}_{j-1})$$ |

where ${b}_{j}\in \{f(x):x\in [{x}_{j-1},{x}_{j}]\}$ is chosen arbitrarily.

If ${b}_{j}=\underset{x\in [{x}_{j-1},{x}_{j}]}{sup}f(x)$, then $S$ is called an *upper Riemann sum*.

If ${b}_{j}=\underset{x\in [{x}_{j-1},{x}_{j}]}{inf}f(x)$, then $S$ is called a *lower Riemann sum*.

For some examples of Riemann sums, see the entry examples of estimating a Riemann integral.

Title | Riemann sum |

Canonical name | RiemannSum |

Date of creation | 2013-03-22 11:49:17 |

Last modified on | 2013-03-22 11:49:17 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 14 |

Author | Wkbj79 (1863) |

Entry type | Definition |

Classification | msc 26A42 |

Related topic | RiemannIntegral |

Related topic | RiemannStieltjesIntegral |

Related topic | LeftHandRule |

Related topic | RightHandRule |

Related topic | MidpointRule |

Defines | left Riemann sum |

Defines | right Riemann sum |

Defines | upper Riemann sum |

Defines | lower Riemann sum |