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

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

Â says. Well here's this statement of the

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

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

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

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

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

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

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

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

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

Â Right? The only thing you really care about is

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

Â You only really want to be able to calculate.

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

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

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

Â fit that single object into a family of objects.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â The derivative of big G is equal to little f.

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

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

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

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

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

Â that big F must be big G plus some constant.

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

Â F of A is equal to 0. Right?

Â This is another fact that I know about big F.

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

Â recover the constant big c. Alright?

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

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

Â Well this means the constant big c must be negative.

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

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

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

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

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

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

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

Â a. And this solves our original problem.

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

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

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

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

Â of problems. Integrating from a to x allowed us to

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

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

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

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

Â And I've got an antiderivative of little f.

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

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

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

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

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

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

Â fundamental theorem of calculus in terms of the accumulation function.

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

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

Â to evaluate integrals. Evaluating an integral is the same as

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

Â endpoints taking the difference.

Â