The Unapologetic Mathematician

Mathematics for the interested outsider

How to Use the FToC

I’ll get back to deconstructing comics another time. For now, I want to push on with some actual mathematics.

After much blood, toil, tears, and sweat, we’ve proven the Fundamental Theorem of Calculus. So what do we do with it? The answer’s in this diagram:

Diagram of the Fundamental Theorem of Calculus

This is sort of schematic rather than something we can interpret literally.

On the left we have real-valued functions — we’re being vague about their domains — and collections of “signed” points. We also have a way of pairing a function with a collection of points: evaluate the function at each point, and then add up all the values or their negatives, depending on the sign of the point. On the right we also have real-valued functions, but now we consider intervals of the real line. We have another way of pairing a function with an interval: integration!

At the top of the diagram, we can take a function and differentiate it to get back another function. At the bottom, we can take an interval and get a collection of signed points by moving to the boundary. The interval \left[a,b\right] has the boundary points \{a^-,b^+\}, where we consider a to be “negatively signed”.

Now, what does the FToC tell us? If we start with a function F in the upper left and an interval \left[a,b\right] in the lower right, we have two ways of trying to pair them off. First, we could take the derivative of F and then integrate it from a to b to get \int_a^b F'(x)dx. On the other hand, we could take the boundary of the interval and add up the function values along the boundary to get F(b)-F(a). The FToC tells us that these two give us the same answer!

To write this in a diagram seems a little much, but keep the diagram in mind. We’ll come back to it later. For now, though, we can use it to understand how to use the FToC to handle integration.

Say we have a function f and an interval \left[a,b\right], and we need to find \int_a^bf(x)dx. We’ve got these big, messy Riemann sums (or Darboux sums), and there’s a lot of work to compute the integral by hand. But notice that the integral is living on the right side of the diagram. If we could move it over to the left, we’d just have to evaluate a function twice and add up the results.

Moving the interval to the left of the diagram is easy: we can just read off the boundary. Moving the function is harder. What we need is to find an antiderivative F(x) so that F'(x)=f(x). Then we move to the left of the diagram by switch attention from f to F. Then we can evaluate F(b)-F(a) and get exactly the same value as the integral we set out to calculate. So if we want to find integrals, we’d better get good at finding antiderivatives!

This has an immediate consequence. Our basic rules of antiderivatives carry over to give some basic rules for integration. In particular, we know that integrals play nicely with sums and scalar multiples:


February 18, 2008 Posted by | Analysis, Calculus | 16 Comments


Okay, usually I’m all behind XKCD, but today’s installment is a bit of a head-scratcher.

The title seems especially ill-chosen. I mean, I know that Randall’s not a doctor of linguistics, but he’s usually pretty on the ball. Clearly he can’t mean the title as a normative statement, but he also has to understand that “how it works” will commonly parse as “how it should work”. The fact that there’s no comeuppance for the jerk doesn’t help here. Without further comment, it’s easy to read the comic as an endorsement of this attitude.

The other thing that leaves a bad taste in my mouth is that the guy on the left is not a clearly-defined character we all know to be unpleasant already. Yes, I know this is arguing semiotics, but there’s a reason Goofus and Gallant comics are so easily read: a generic character will be interpreted as a generic person. Their behavior is then also taken as generic. Putting the Hat Guy in there would go a long way towards making this not seem like an endorsement.

And then the details are off. The characters are looking at a calculus problem. I don’t know anyone — at least any instructor — in this day and age who thinks like this at the calculus level. As far as I know, the psychological damage is usually done by this point. The attitude comes in during grade-school, so an arithmetic problem (and younger characters at the board) would be more appropriate. That is, unless Randall is asserting that this attitude is endemic (remember generic character => generic person) among calculus instructors.

In that case I really have to disagree with him on the strongest possible terms. But again, there’s no further comment, and the whole thing just feels disappointing as a result.

February 18, 2008 Posted by | rants | 43 Comments