[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.