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}](http://l.wordpress.com/latex.php?latex=f%3A%5Cleft%5Ba%2Cb%5Cright%5D%5Crightarrow%5Cmathbb%7BR%7D&bg=e6e6e6&fg=000000&s=0 "f:\left[a,b\right]\rightarrow\mathbb{R}")
![c\in\left[a,b\right]](http://l.wordpress.com/latex.php?latex=c%5Cin%5Cleft%5Ba%2Cb%5Cright%5D&bg=e6e6e6&fg=000000&s=0 "c\in\left[a,b\right]")
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 
![\left[a,b\right]](http://l.wordpress.com/latex.php?latex=%5Cleft%5Ba%2Cb%5Cright%5D&bg=e6e6e6&fg=000000&s=0 "\left[a,b\right]")
Now the Intermediate Value Theorem tells us that
just as we wanted.
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This theorem is just a reformulation of the differential MVT if you take into account the fundamental theorem of calculus (exercise).
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.
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Great post! Very powerful theorems.