Okay, last time we talked about the

Second Fundamental Theorem of Calculus. That said that if big F is

an antiderivative of little f, in other words, if the derivative

of F is f, then computing definite integrals

of f is easy. To get the definite integral from

a to b, you just apply F at the end points. That’s really useful because the

definition of this is the limit of a sum and it can get really, really ugly. Finding an antiderivative and plugging

it in is much, much easier. But that brings up a question. How do you know that there

is an antiderivative? And if there is, how can you find it? That brings us to the First Fundamental

Theorem of Calculus. The First Fundamental Theorem of Calculus

says that every continuous function has an antiderivative. And in fact, the indefinite integral

is that antiderivative. It says if you compute the indefinite

integral, and then take the derivative of the indefinite integral, you get back

the original function. Let’s look at a picture of

what’s going on. Let’s suppose we’ve got our function f. Here’s our function, y=f(s). Here’s our variable s. We start at a certain point a,

and then we say we’ll define the function i(x) is the

integral from a to x of f(s)ds. In other words, if you pick a value x,

then i(x) is all the area under the curve between a and x. Okay, great! It’s a function. If you changed the value of that function,

you make x a little bit bigger, you pick up some extra area.

You pick up this region. If you make x smaller,

you lose some area. So i(x) is the total accumulated area

as you sweep out, starting from a until you get to x. Let’s go ahead and see if we can take

the derivative of this. We’re not gonna use any of our

derivative formulas like the derivative of x^3 or the chain rule or the

product rule or the quotient rule. We’re gonna go back to the definition

of the derivative. The derivative of i(x) is the limit as

h goes to 0 of [ i (x+h) – i(x) ] / h. Let’s draw the point (x+h)

and see what is i(x+h)? i(x+h) is all the area

from a to (x+h). And i(x) is all the area from a to x. The difference is all of the area

between x and (x+h). In other words, it’s the integral

from x to (x+h) of f(s)ds, and we’re dividing everything by h. How big is this area? Well it’s roughly a rectangle. A little shorter than that rectangle,

a little bigger than that rectangle. But it’s roughly a rectangle,

and the width of that rectangle is h. So if you take the area of this region

and divide by h, what you’re really doing is taking the average height. And you’re averaging between x and (x+h). The height is not exactly constant.

Here, it’s a little bit bigger than here. But if you take a very small value of h,

and you look only in this really small region, then the height doesn’t

change much at all. In fact, as long as a

function is continuous, as you make this thinner and thinner

and thinner and thinner, you get less and less wiggle room

about how high the function is. So the limit is the actual height

at the point x. In other words, the limit is f(x). We have just proved that half of

the Fundamental Theorem of Calculus. What’s the big deal?

Now that we’ve proved it, what can you do with it? One thing is we’ve got three different

notions, remember? We have the definite integral. The definite integral is the

integral from a to b of f(x)dx. Which is the same as the integral

from a to b of f(s)ds, doesn’t matter what we call the variable. We have our indefinite integral, which is

the integral from a to x of f(s)ds. And we’ve got our antiderivative,

which we called F(x). Now the point is we just showed that

the indefinite integral is an antiderivative. For that matter, any antiderivative

is an indefinite integral. All antiderivatives are the same. They’re all the same up to a constant. The same way all indefinite integrals

are the same. See, if we changed the value of a, if we started a over here,

then we would pick up some extra area. That would add a constant to i(x). Different choices of ‘a’ are like

adding constants to f(x). In fact, the notation that people

often use, they would use integral of f(x)dx and they

will call this an indefinite integral. But usually what they mean is that

they’re looking for an antiderivative. It’s bad terminology. An indefinite integral really means this,

but when you see this notation in books, as often as not, people are just

talking about antiderivatives because we’ve just shown indefinite integrals

are antiderivatives. Antiderivatives are indefinite integrals.