# example of integral mean value theorem

###### Example.

If $f$ is a continuous^{} real function on an interval $[a,b]$, then there exists a $\zeta \in (a,b)$ such that

$${\int}_{a}^{b}f(x)dx=f(\zeta )(b-a).$$ |

###### Proof.

Let $g(x)\equiv 1$. Then by the Integral Mean Value Theorem, there exists $\zeta \in (a,b)$ such that

${\int}_{a}^{b}}f(x)dx$ | $={\displaystyle {\int}_{a}^{b}}f(x)g(x)dx$ | ||

$=f(\zeta ){\displaystyle {\int}_{a}^{b}}g(x)dx$ | |||

$=f(\zeta ){\displaystyle {\int}_{a}^{b}}1dx$ | |||

$=f(\zeta )(b-a)$ |

as required. ∎

Title | example of integral mean value theorem |
---|---|

Canonical name | ExampleOfIntegralMeanValueTheorem |

Date of creation | 2013-03-22 18:20:24 |

Last modified on | 2013-03-22 18:20:24 |

Owner | me_and (17092) |

Last modified by | me_and (17092) |

Numerical id | 4 |

Author | me_and (17092) |

Entry type | Example |

Classification | msc 26A06 |