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[MUSIC]. 

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How can I use the fundamental theorem of 
of calculus to evaluate integrals? 

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Well let's remember back to what the 
fundamental theorem of calculus actually 

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says. 
Well here's this statement of the 

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fundamental theorem of calculus. 
I got that function little from the 

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closed interval to the real numbers and 
it's continuous. 

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And let the function big F, which is the 
accumulation function, it's the interval 

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from a to x of f of t dt. 
That's the definition of f of x. 

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And then big F is continuous on the 
closed interval, differentiable on the 

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open interval, and the derivative of big 
f is a little f. 

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In other words, an antiderivative for 
little f, is big f. 

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What is it that I want to calculate? 
What you don't really want to calculate 

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is th, is this. 
I mean, who really cares about big f? 

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Right? 
The only thing you really care about is 

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the integral from a to be of f of t dt, 
right? 

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You only really want to be able to 
calculate. 

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Big F of b. 
Who cares about big F of x? 

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This sort of trick comes up all of the 
time in mathematics. 

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To understand an object in isolation, to 
understand big F of b, it's necessary to 

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fit that single object into a family of 
objects. 

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In this particular case, you're right. 
I don't care about big F of x. 

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But I wanta calculate big F of b. 
I, I wanta be able to integrate from a to 

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b, f of t, dt. 
and by fitting F of b into this function 

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of F of x, I can then try to understand 
something about F of b because I know how 

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big F changes. 
I know that the derivative of big F. 

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Is little f. 
So what do I know about the accumulation 

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function, big F of x. 
Well, I know that big F of a is equal to 

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0, because it's the integral from a to a 
of f of t d t. 

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So I know the value of big F at a And I 
know how F changes. 

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I know that the derivative of F is the f. 
And now I'm trying to calculate F of b. 

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So I know that F of a is equal to 0. 
And I know something about how F changes. 

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The change in F is related to F. 
That is enough information to recover big 

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F. 
So before we recover big F let's try to 

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walk there in steps. 
Let's first supposed that I've got some 

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function big G, which is some anti 
derivative of little f. 

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Right. 
So in other words, I mean that... 

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The derivative of big G is equal to 
little f. 

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Well, how does big G compare to big F? 
Well, I also know that big F 

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differentiates to little f, right, so I 
know that big G's derivative is little f, 

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and that's the same as the derivative of 
big F. 

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So I've got 2 functions who's derivative 
is the same. 

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What does that tell me? 
Well by the mean value theory that means 

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that big F must be big G plus some 
constant. 

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What's the constant? 
Well, the other fact that I know is that 

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F of A is equal to 0. 
Right? 

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This is another fact that I know about 
big F. 

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And this fact, and the fact that big f of 
x is big g of x plus c is enough to 

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recover the constant big c. 
Alright? 

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If big f of a is equal to 0 that's also g 
of a plus c. 

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So what does c have to be so that if I 
add it to g of a I get 0? 

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Well this means the constant big c must 
be negative. 

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G of a. 
Well now we can put it all together. 

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This fact and this fact then combine to 
give me that big f of x is big g of x, 

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minus big g of a. 
So does that help at all? 

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Yes, this solves all of our problems. 
Remember what big f of x was. 

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Big F of x was the integral from a to x 
of f of t d to. 

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And what this formula's telling you is 
that this intregal is some 

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anti-derivative for little f, evaluated x 
minus that anti-derivative evaluated at 

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a. 
And this solves our original problem. 

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The original problem that we wanted to 
answer was just the integral from a to b 

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of f of t d t. 
And that by substituting in b for x, is 

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big g of b minus big g of a. 
But we wouldn't have been able to 

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understand this had I not fit this 
specific problem into a whole collection 

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of problems. 
Integrating from a to x allowed us to 

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understand this particular integral from 
a to a specific value, b. 

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Let's change the names around and 
summarize what we've got. 

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So here's how this is usually summarized. 
Suppose I've got sum function little f 

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from the closed interval a to b to the 
real numbers. 

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And I've got an antiderivative of little 
f. 

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The here I'm calling big F but before we 
were calling it big G. 

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Anyhow, then what do we know? 
Then the integral from a to b of little f 

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which you remember before I was calling 
it little f of t dt just so I didn't 

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confuse my variables but I can call it x, 
because I don't have any t's here. 

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The integral from a to b of f of x d x is 
that antiderivative evaluated at b minus 

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that antiderivative evaluated at a. 
We started off with a formulation of the 

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fundamental theorem of calculus in terms 
of the accumulation function. 

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But now we've got this new formulation of 
the fundamental theorem of calculus, 

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makes it a lot easier to see how we can 
use the fundamental theorem of calculus 

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to evaluate integrals. 
Evaluating an integral is the same as 

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finding an antiderivative and evaluating 
that antiderivative at left and the right 

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endpoints taking the difference.