Concept: Mean Value Theorem for Integrals
Recall from differential calculus that the Mean Value Theorem for Derivatives stated that a continuous function on a given closed interval must contain a point such that the derivative of the function at that point is the average derivative on that interval.
Okay, now shake out the confusion from your head. I understand that stating the Mean Value Theorem this way, without the symbols and function notation, is confusing; however, you need to start conceptualizing the stuff you have learned to understand what it really means. The best way to do this is to avoid symbolic notation, and instead focus on what the gist of the theorem is.
Below is a Wolfram Demonstrations app that should help you recall the Mean Value Theorem for Derivatives.
Okay, now shake out the confusion from your head. I understand that stating the Mean Value Theorem this way, without the symbols and function notation, is confusing; however, you need to start conceptualizing the stuff you have learned to understand what it really means. The best way to do this is to avoid symbolic notation, and instead focus on what the gist of the theorem is.
Below is a Wolfram Demonstrations app that should help you recall the Mean Value Theorem for Derivatives.
Mean Value Theorem from the Wolfram Demonstrations Project by Michael Trott
Now that we know how to find the average value of a function on a given, closed interval, it would be incredibly nice to know where this average value occurs. That's where the next theorem (Mean Value Theorem for Integrals) comes in. It essentially states that, given a continuous function on a closed interval (see the similarity with the MVT for Derivatives so far?), there must exist a point in the interval such that the value of the function at that point is the average value on that interval. Thus, some function values on the interval will likely be greater than this value, and some will likely be less than it, but in the end this guy is the average!
"Isn't this just the average value of a continuous function theorem you just mentioned?"
That's a good question, voice in my head. The theorem above concerning the average value of a function just says that we can find the average value of a continuous function on a closed interval, but it says nothing about the fact that this average will happen on the closed interval itself!
Let me be clear! The average value of a continuous function theorem gives us the tools to find the average value of a function. The Mean Value Theorem for Integrals tells us that this average value occurs at some value of \(x \in [a,b]\). The difference is subtle when first learning this concept, but not too terribly difficult to grasp.
Before going deeper into the Mean Value Theorem for Integrals, let's try to visualize what it is saying. Below we have a function graphed for us. You can change the left and righthand points of the closed interval if you want. You can also rescale the graph for a better visual effect. Notice that no matter how you place the left and righthand points for the closed interval, the average value of the function (indicated by the horizontal line) is between the lowest function value and the highest function value on that interval.
"Isn't this just the average value of a continuous function theorem you just mentioned?"
That's a good question, voice in my head. The theorem above concerning the average value of a function just says that we can find the average value of a continuous function on a closed interval, but it says nothing about the fact that this average will happen on the closed interval itself!
Let me be clear! The average value of a continuous function theorem gives us the tools to find the average value of a function. The Mean Value Theorem for Integrals tells us that this average value occurs at some value of \(x \in [a,b]\). The difference is subtle when first learning this concept, but not too terribly difficult to grasp.
Before going deeper into the Mean Value Theorem for Integrals, let's try to visualize what it is saying. Below we have a function graphed for us. You can change the left and righthand points of the closed interval if you want. You can also rescale the graph for a better visual effect. Notice that no matter how you place the left and righthand points for the closed interval, the average value of the function (indicated by the horizontal line) is between the lowest function value and the highest function value on that interval.
Integral Mean Value Theorem from the Wolfram Demonstrations Project by Chris Boucher
Another important concept to think about is that, if that horizontal line represents the average height on the interval, then we could use the area of the resulting rectangular region created by this horizontal averaging line to compute the integral of \(f(x)\) on that interval. That is, instead of computing the integral from \(a\) to \(b\), we could just find the average height and multiply it by the distance \(b  a\). This is akin to saying that you're in a class of 6 students and when you guys compare the amount of money each of you has in your wallets, yours ends up being the exact average. Rather than adding up everyone's amounts to find the total money in the room, you could just multiply your amount (the representative exact average) by 6.

