The Integral Mean Value Theorem « 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 John Armstrong | Analysis, Calculus | | 5 Comments

5 Comments »

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 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
[...] 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 [...]

Pingback by The Fundamental Theorem of Calculus I « The Unapologetic Mathematician | February 13, 2008
[...] 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

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