This article originated from personal public account: TechFlow, original is not easy, for attention


Today, the 12th installment of our advanced mathematics series, we’ll continue with definite integrals.

When WE talked about derivatives, we introduced a series of derivations of the mean value theorem of derivatives. So if we have the mean value theorem of differentiation, then we also have the mean value theorem of integration, so let’s look at the definition of the mean value theorem of integration.

Extremum theorem

The extremum theorem is also called the maximum and minimum value theorem, and its meaning is very intuitive: if the function f(x) is a continuous function on the interval [a, B], there must be a maximum and minimum value, and the maximum and minimum value are taken at least once.

This is a very famous theorem, the content of the theorem is very intuitive, and not difficult to understand. However, it is not easy to prove it, which is derived from several theorems such as interval sleeve theorem and B-M theorem. This section of proof process is quite complicated, and due to the limitation of space and level, this part can only be skipped in this paper, and interested students can understand it by themselves.

We assume that m and m are the minimum and maximum values of the function f(x) on the interval [a, b] respectively, then according to the extremum theorem, the following formula can be obtained:


This might look a little complicated, but when we draw it, it’s very simple:


The gray shaded area in the figure above is the definite integral, the blue rectangle is m(b-a), and the large rectangle is m(b-A).

We can prove this very easily by looking at geometric areas.

And the math is easy, because m is the minimum and m is the maximum, so we can get. We take the constant as a function and integrate it, so we can get:


The integral of both sides is going to be the area of the rectangle, and then we have the proof.

Mean value theorem of integrals

The extremum theorem is very simple, but it’s the basis of a lot of theorems, like our mean value theorem for integrals.

So let’s just do a simple transformation of the top one, and since b minus a is constant and greater than 0, we have b minus aDivide both sides of this inequality by b-a, and you get:


We put theThis is the whole thing, and it’s somewhere between the maximum and minimum of the function. According to theIntermediate value theorem for continuous functionsWe can certainly find something on [a, b], such that f of x is atThe value at this point is the same as this value, which means:


So this is the mean value theorem for integrals, and there are two things to notice here, and let’s start with the easy one, which is that we use the continuous function intermediate value theorem. So this is bound to be oneContinuous functionOtherwise, the function might happen to be inThere’s no definition at point. This is also a prerequisite for the theorem to be true.

The second point is a brief introduction to the intermediate value theorem of continuous functions, which means that for a continuous function on the interval [a, b], for any constant between its maximum value and minimum value, we must be able to find a point on the interval [A, b] where the value of the function is equal to the constant.


With those details in mind, let’s look at the formula:


Let’s multiply the constants back:


What’s the integral on the right hand side? It’s the area of the curve of the function, but now we’ve converted it to a function times the width, so we can view it as the height of the rectangle. Let’s look at the picture below.


That is to sayThe area of the rectangle with height is the same as the area of the curve enclosed by the function, so it is both the height of the rectangle and really the average of the function on [a, b].

conclusion

The mean value theorem is one of the most important theorems in the field of calculus. It is also the context of the whole calculus. We’re familiar with the derivation of the mean value theorem, and it’s very helpful to deepen our understanding of calculus. More importantly, the derivation of both theorems is relatively easy and interesting, so it is recommended that you try them yourself.

Today’s article is all of these, if you feel that there is a harvest, please feel free to point attention or forward, your help is very important to me.