# The Unapologetic Mathematician

## 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 $f:\left[a,b\right]\rightarrow\mathbb{R}$ then there is some $c\in\left[a,b\right]$ so that

$\displaystyle f(c)=\frac{1}{b-a}\int\limits_a^bf(x)dx$

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 $a$ and $b$. Then the upper Darboux sum is $(b-a)M$, where $M$ is the maximum value of $f$ on the interval $\left[a,b\right]$. Similarly, the lower Darboux sum on this interval is $(b-a)m$ (where $m$ is the minimum value of $f$), and it’s the lowest possible Darboux sum. Then we can divide everything in sight by $b-a$ to get the inequality

$\displaystyle m\leq\frac{1}{b-a}\int\limits_a^bf(x)dx\leq M$

Now the Intermediate Value Theorem tells us that $f$ must take every value between $m$ and $M$ at some point between $a$ and $b$. And thus there must exist a $c\in\left[a,b\right]$ so that

$\displaystyle f(c)=\frac{1}{b-a}\int\limits_a^bf(x)dx$

just as we wanted.

February 12, 2008 - Posted by | Analysis, Calculus

1. 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 | Reply

2. Which theorem I have not yet covered. Patience, Michael. I know you know far better than I how I should be 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 | Reply

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

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5. Great post! Very powerful theorems.

Comment by Daily Calculus | April 9, 2008 | Reply

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

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7. […] 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 […]

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

Comment by Robert | May 3, 2017 | Reply