[MUSIC].
How can I use the fundamental theorem of of calculus to evaluate integrals?
Well let's remember back to what the fundamental theorem of calculus actually
says. Well here's this statement of the
fundamental theorem of calculus. I got that function little from the
closed interval to the real numbers and it's continuous.
And let the function big F, which is the accumulation function, it's the interval
from a to x of f of t dt. That's the definition of f of x.
And then big F is continuous on the closed interval, differentiable on the
open interval, and the derivative of big f is a little f.
In other words, an antiderivative for little f, is big f.
What is it that I want to calculate? What you don't really want to calculate
is th, is this. I mean, who really cares about big f?
Right? The only thing you really care about is
the integral from a to be of f of t dt, right?
You only really want to be able to calculate.
Big F of b. Who cares about big F of x?
This sort of trick comes up all of the time in mathematics.
To understand an object in isolation, to understand big F of b, it's necessary to
fit that single object into a family of objects.
In this particular case, you're right. I don't care about big F of x.
But I wanta calculate big F of b. I, I wanta be able to integrate from a to
b, f of t, dt. and by fitting F of b into this function
of F of x, I can then try to understand something about F of b because I know how
big F changes. I know that the derivative of big F.
Is little f. So what do I know about the accumulation
function, big F of x. Well, I know that big F of a is equal to
0, because it's the integral from a to a of f of t d t.
So I know the value of big F at a And I know how F changes.
I know that the derivative of F is the f. And now I'm trying to calculate F of b.
So I know that F of a is equal to 0. And I know something about how F changes.
The change in F is related to F. That is enough information to recover big
F. So before we recover big F let's try to
walk there in steps. Let's first supposed that I've got some
function big G, which is some anti derivative of little f.
Right. So in other words, I mean that...
The derivative of big G is equal to little f.
Well, how does big G compare to big F? Well, I also know that big F
differentiates to little f, right, so I know that big G's derivative is little f,
and that's the same as the derivative of big F.
So I've got 2 functions who's derivative is the same.
What does that tell me? Well by the mean value theory that means
that big F must be big G plus some constant.
What's the constant? Well, the other fact that I know is that
F of A is equal to 0. Right?
This is another fact that I know about big F.
And this fact, and the fact that big f of x is big g of x plus c is enough to
recover the constant big c. Alright?
If big f of a is equal to 0 that's also g of a plus c.
So what does c have to be so that if I add it to g of a I get 0?
Well this means the constant big c must be negative.
G of a. Well now we can put it all together.
This fact and this fact then combine to give me that big f of x is big g of x,
minus big g of a. So does that help at all?
Yes, this solves all of our problems. Remember what big f of x was.
Big F of x was the integral from a to x of f of t d to.
And what this formula's telling you is that this intregal is some
anti-derivative for little f, evaluated x minus that anti-derivative evaluated at
a. And this solves our original problem.
The original problem that we wanted to answer was just the integral from a to b
of f of t d t. And that by substituting in b for x, is
big g of b minus big g of a. But we wouldn't have been able to
understand this had I not fit this specific problem into a whole collection
of problems. Integrating from a to x allowed us to
understand this particular integral from a to a specific value, b.
Let's change the names around and summarize what we've got.
So here's how this is usually summarized. Suppose I've got sum function little f
from the closed interval a to b to the real numbers.
And I've got an antiderivative of little f.
The here I'm calling big F but before we were calling it big G.
Anyhow, then what do we know? Then the integral from a to b of little f
which you remember before I was calling it little f of t dt just so I didn't
confuse my variables but I can call it x, because I don't have any t's here.
The integral from a to b of f of x d x is that antiderivative evaluated at b minus
that antiderivative evaluated at a. We started off with a formulation of the
fundamental theorem of calculus in terms of the accumulation function.
But now we've got this new formulation of the fundamental theorem of calculus,
makes it a lot easier to see how we can use the fundamental theorem of calculus
to evaluate integrals. Evaluating an integral is the same as
finding an antiderivative and evaluating that antiderivative at left and the right
endpoints taking the difference.