## The Integral Mean Value Theorem

Okay: time to get back on track. Today, we’ll see a theorem about integrals that’s similar to the Differential Mean Value Theorem. Specifically, it states that if we have a continuous function then there is some so that

Let’s consider the Darboux sums we use to define the integral. We know that if we choose a partition, then its upper Darboux sum is greater than any Riemann sum of any refinement of that partition. So let’s take the absolute coarsest possible partition: the one where we just have partition points and . Then the upper Darboux sum is , where is the maximum value of on the interval . Similarly, the lower Darboux sum on this interval is (where is the minimum value of ), and it’s the lowest possible Darboux sum. Then we can divide everything in sight by to get the inequality

Now the Intermediate Value Theorem tells us that must take every value between and at some point between and . And thus there must exist a so that

just as we wanted.

This theorem is just a reformulation of the differential MVT if you take into account the fundamental theorem of calculus (exercise).

Comment by Michael Livshits | February 13, 2008 |

Which theorem I have not yet covered. Patience, Michael. I know you know far better than I how I

shouldbe covering this material, but in the absence of your definitive weblog on the subject, I’m going through it in my own way.Comment by John Armstrong | February 13, 2008 |

[…] now let’s use the Integral Mean Value Theorem to get at the integral here. It tells us that there’s some between and with — the […]

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[…] Mean Value Theorems We’ve got two different analogues of the integral mean value theorem for the Riemann-Stieltjes […]

Pingback by Two Mean Value Theorems « The Unapologetic Mathematician | March 28, 2008 |

Great post! Very powerful theorems.

Comment by Daily Calculus | April 9, 2008 |

[…] Mean Value Theorem for Multiple Integrals As in the single variable case, multiple integrals satisfy a mean value […]

Pingback by The Mean Value Theorem for Multiple Integrals « The Unapologetic Mathematician | December 29, 2009 |

[…] Integral Mean Value Theorem We have an analogue of the integral mean value theorem that holds not just for single integrals, not just for multiple integrals, but for integrals over […]

Pingback by The Integral Mean Value Theorem « The Unapologetic Mathematician | June 14, 2010 |

How can I get the version of mean value theorem for general measure space

Comment by Robert | May 3, 2017 |

Here you go, Robert: https://unapologetic.wordpress.com/2010/06/14/the-integral-mean-value-theorem-2/

Comment by John Armstrong | May 3, 2017 |