The Intuitive Proof of The Fundamental Theorem of Calculus
I hope that you've found the FTC at least a little bit surprising, and I further hope that want to get a sense for "why" the thing is true. Let's talk about that.
In order to give an intuitive proof of the FTC, we need an explanation of why the area on the left and the line segment on the right of the applets in the previous activity are equal. The easiest way to understand this is in the context of a real-world application.
To that end, let's get back to the model of the traffic on Route 15 in Johnson Vermont. The applet below is just like the two from the previous activity, but the function (on the left) to be integrated is now
g(x), the model of the rate of traffic on Route 15 in Johnson Vermont. As we saw earlier, integrals of g(x) represent the total car count between two periods of time. For instance, the integral of g(x) between minute a=360 (6am) and minute b=960 (4pm) is about 5150, and this number represents the total car count on Route 15 in this time period.
Don't worry about the right panel with G(x) just yet. For now, adjust a and b, refreshing your memory of the significance of this integral. If you need it, the "reset" button is in the top right of the left panel.Let's turn our attention to the function
G(x) in the right panel. First of all, either check or trust me that G(x) is an antiderivative of g(x). If you want to check, you'll need to calculate G'(x) and see that it is equal to g(x). I don't want to get bogged down in that Monkey Rule calculation however, since it takes our focus off the main event: understanding why the FTC is true.
Now, take a moment and ponder this question: What real world quantity does the antiderivative G(x) model? Keep in mind, its derivative G'(x) is equal to g(x), and g(x) is a model of the rate of traffic on Route 15.
If you aren't quite sure about that, let me ask you a leading question: What must the units of G(x) and x be so that the derivative G'(x) (which is equal to g(x)) will have units "cars-per-minute"?
According to the rule about the units of the derivative, in order for G'(x) to have units "cars-per-minute", G(x) must have units "cars", and x must have units "minutes".
Do you see where this is going? We just learned that the units of G(x) must be "cars" and the units of x must be "minutes", and the derivative of G(x) is a model of the rate of traffic on the highway. That only leaves one possibility for G(x): it must be a model of the car count on Route 15. We've come to this model of car count in a roundabout way, but it's the only possibility for G(x). The mathematical way of talking about G(x) is that it is a model of the accumulated effect of the rate modeled by g(x).
But what time does the model "start at"? In other words, what does G(0) represent? This is where the negative 572.31312 comes in at the end of G(x). Notice, because of Monkey Rule 2, there can be any number at the end of the G(x) and its derivative will still be g(x). This is called the "Constant of Integration". I carefully selected negative 572.31312 in order to make G(0) equal to 0. By doing this, I've forced G(x) to be a model of car count between midnight and minute x. We'll take a closer look at the constant of integration in the next activity.
For now though, let's take another look at the right panel in the applet above. We now know that the antiderivative G(x) is a model of total car count on Route 15 since midnight. So G(0) is 0 (because no cars drive past in no minutes), G(960) is the total number of cars that have travelled on the highway between midnight and 4pm, and G(360) is the total number of cars that have travelled on the highway between midnight and 6am.
Therefore G(960)-G(360) must represent the total number of cars that travelled along the highway between 6am and 4pm. Indeed, using just arithmetic and common sense:
(Traffic between midnight and 4pm) - (Traffic between midnight and 6 am)
=
(Traffic between 6am and 4pm)
Wait a second though. "Traffic between 6am and 4pm" is exactly what we were trying to calculate with the integral on the left back at the start of this chapter when we started studying integrals with rectangles.
And this is the core of the intuitive proof of the Fundamental Theorem of Calculus. You can calculate the total car count on Route 15 either by calculating the area under the graph of g(x) or by taking the difference of two readings from G(x). Both represent the same quantity, and for that reason, both must be equal. In other words, so long as G'(x) is equal to g(x):
Before you move on, I want to emphasize that this is not a mathematical proof of the FTC. It is only an intuitive confirmation that the FTC works. In order to prove the FTC one would need to confirm the mathematical statement is true independent of any mathematical model. We will not undertake that here since our main goal is applications.
In the next activity, we'll discuss the constant of integration (negative 572.31312) that appeared in G(x) somewhat mysteriously above.