Welcome to calculus. I'm professor Ghrist.

We're about to begin lecture 26, bonus material.

In our main lesson, we covered the fundamental theorem of integral calculus.

We saw how it was used, and what it meant.

But we did not say why it is true. Let's sketch a proof, but how do we

proceed? Our goal is to show that the definite

integral of f from a to b is the indefinite integral evaluated at the

limits. We won't prove that directly rather, we

will prove a lemma, a preparatory step. The lemma is going to look very

different, but is really closely related. It states the following, the integral of

f of t dt, as t goes from a to x is what? Well, let's see, this is going to be a

function of x. If we differentiate that with respect to

x, what will we get? We will get f of x, the integrand

evaluated at x. Now this seems a little unusual since

you're differentiating a limit of a definite integral, but lets explore, see

if we can make sense of it. Lets make sure we're not completely crazy

and check that this works in a simple case.

Lets differentiate the integral as t goes from a to x of a constant dt.

Well, we can do the definite integral of a constant, c.

And that's just going to give us c times the upper limit x minus c times the lower

limit, a. If we differentiate that function of x,

what do we get? We get c, the constant which is the

integrand we began with. So, this is not a clearly crazy thing to

do. Well, let's see if we can prove this

lemma. Consider what the integrand f of t looks

like, we need to compute the definite integral from a to x.

And the claim is that the derivative of the definite integral is f evaluated at

x. Well, lets see, let's denote by capital

F, the definite integral as a function of x.

Then, if we want to differentiate that with respect to x, what should we do?

Let's look at what happens when we increase x by a small amount.

Let's call that h. Well, according to the definition, that

is the integral of f of t dt, as t goes from a to x plus h.

Now of course, we could write that as the integral as t goes from a to x, plus the

integral as t goes from x to x plus h. That comes from additivity of the

integral. And since by definition, the integral

from a to x is simply capital F at x, we see something that should start looking

like the definition of a derivative. Namely capital F at x plus h equals

capital F at x plus something. What is that something?

That something is going to have the derivative of capital F built into it.

Now, what we need to focus on is this interval from x to x plus h.

And here, I'm going to I have to make a little bit of a more restrictive

assumption about the integrand f namely, that not only is it continuous but it

also is reasonable. lets say has a a Taylor series associated

to it. Now, for this definite integral, lets

choose a partition with width h. What is the height?

Well, the height if we choose the left hand endpoint would be f of x.

The width is h now, this is not exactly what the definite integral is and it's

just an approximation, there's some leftover stuff that we haven't accounted

for. How can we estimate that?

Well, if the intagrand f has a a reasonable form to it, let's say it has a

well defined derivative in the sense of a Taylor series.