The Magic Theorem, Part III

The previous two posts have wrestled with the relationship between an area (the “definite integral”) and antiderivative (the “indefinite integral”). In short, they are related by the Fundamental Theorem, or as I have been calling it, the Magic Theorem:

*You can find the area under a graph by finding how much its
antiderivative has changed.*

In mathematical notation:

But that still leaves us with the question:

*Why do we care about area in physics?*

I think the answer to that question is that really, we very rarely care about the literal “area” as in area of some planar region, measured in square meters. But we frequently care about other physical values that we can represent as areas. And the methods we use to find those areas can be extended to other useful things.

To explain this, in this post we will step back and look at
how you can find area in math class *before*
you learn the Magic Theorem.

*[Just for extra emphasis, I will say again: what follows is not a method you will need in AP Physics. In fact, except for a few days, you won’t even use this method in math class either. But if you want to understand what the Magic Theorem does for you, you have to understand what it replaces.]*

In math class one day, your teacher might ask you to find the area between the x-axis and the function *f(x)=x ^{2}*:

Now maybe rectangle areas are the only ones you know:

area = length times width

You are kind of stuck. But your teacher asks you to keep trying and says they are willing to accept an *approximate* answer. Oooh. Now there is a way forward here.

You could slice the region into a handful of rectangles. Let’s start with five of them:

You could then calculate the area of those rectangles and add them up to get a rough idea of the area of the entire region. [Notice, I say we COULD do this. It’s a tedious task to do by hand, and in the end we won’t have to do it. But we could if we wanted to. As you can see, I let Desmos do the calculations for us.]

In the diagram above, the rectangles are INSIDE the region, so we have underestimated the area. We could have chosen to put them OUTSIDE, thus overestimating:

So we can see that the area we are looking for is more than 30 and less than 55. That’s an approximation, but it’s better than no knowledge at all. It might even satisfy your math teacher for a while. But then they come back to you with a request:

“I like your method, but I wish it were more precise. Can you divide your graph into more, finer intervals? Like maybe 10 of them? Or even 50?

Well, you can do this too. Here are 10 “outside” rectangles:

And here, side by side, we have 50 “inside” rectangles and 50 “outside”:

We can now make two observations about what happens when you use lots of rectangles:

1. The difference between “inside” and “outside” estimates becomes smaller and smaller.

2. The amount that you are overestimating or underestimating by also gets smaller and smaller.

If we could some how let the
number of rectangles approach infinity, we would know the true area of this
region. In fact, that limit is the *definition* of the area of this region.

This way of finding area is called the Method of Reimann Sums. The area is often expressed using summation notation:

Setting up these expressions and then calculating the limit as the number of rectangles approaches infinity can be quite daunting. Mastering this method is a rite of passage in calculus class and I admit that it is possible you will not enjoy the process. It is good to understand what Riemann Sums are, but it is also really nice to not have to use them. That’s what is so magical about the Fundamental Theorem — it is so much easier that it feels like magic. Just to remind you of what we do:

1. We find a function which
has x^{2} as its derivative. For
example:

2. According to our magic theorem, the area we want is the change in that antiderivative:

That’s not so bad! But also, now we have all the pieces in place to answer the question that started us down this road: why do we care about area in physics?

Onward to Part IV of our adventure!