# left hand rule

The *left hand rule* for computing the Riemann integral $\underset{a}{\overset{b}{\int}}}f(x)\mathit{d}x$ is

$$\underset{a}{\overset{b}{\int}}f(x)\mathit{d}x=\underset{n\to \mathrm{\infty}}{lim}\sum _{j=1}^{n}f\left(a+(j-1)\left(\frac{b-a}{n}\right)\right)\left(\frac{b-a}{n}\right).$$ |

If the Riemann integral is considered as a measure^{} of area under a curve, then the expressions $f\left(a+(j-1)\left({\displaystyle \frac{b-a}{n}}\right)\right)$ the of the rectangles, and $\frac{b-a}{n}$ is the common of the rectangles.

The Riemann integral can be approximated by using a definite value for $n$ rather than taking a limit. In this case, the partition^{} is $\{[a,a+{\displaystyle \frac{b-a}{n}}),\mathrm{\dots},[a+{\displaystyle \frac{(b-a)(n-1)}{n}},b]\}$, and the function is evaluated at the left endpoints of each of these intervals. Note that this is a special case of a left Riemann sum in which the ${x}_{j}$’s are evenly spaced.

Title | left hand rule |

Canonical name | LeftHandRule |

Date of creation | 2013-03-22 15:57:38 |

Last modified on | 2013-03-22 15:57:38 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 16 |

Author | Wkbj79 (1863) |

Entry type | Theorem |

Classification | msc 41-01 |

Classification | msc 28-00 |

Classification | msc 26A42 |

Related topic | RightHandRule |

Related topic | MidpointRule |

Related topic | RiemannSum |

Related topic | ExampleOfEstimatingARiemannIntegral |