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## The fundamental theorem of calculus and definite integrals

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# The fundamental theorem of calculus and definite integrals

AP Calc: FUN‑6 (EU), FUN‑6.B (LO), FUN‑6.B.1 (EK), FUN‑6.B.2 (EK), FUN‑6.B.3 (EK)

## Video transcript

Let's say we've
got some function f that is continuous over
the interval between c and d. And the reason why I'm using
c and d instead of a and b is so I can use a
and b for later. And let's say we set up
some function capital F of x which is
defined as the area under the curve between
c and some value x, where x is in this interval
where f is continuous, under the curve-- so it's the
area under the curve between c and x-- so if this is x right
over here-- under the curve f of t dt. So this right over here,
F of x, is that area. That right over there
is what F of x is. Now, the fundamental
theorem of calculus tells us that if f is continuous
over this interval, then F of x is differentiable at
every x in the interval, and the derivative of capital
F of x-- and let me be clear. Capital F of x is differentiable
at every possible x between c and d, and the
derivative of capital F of x is going to be equal
to lowercase f of x. Fair enough. Now, what I want
to do in this video is connect the first
fundamental theorem of calculus to the second part, or the
second fundamental theorem of calculus, which we tend
to use to actually evaluate definite integrals. So let's think about what F of
b minus F of a is, what this is, where both b and a are
also in this interval. So F of b-- and we're going to
assume that b is larger than a. So let's say that b is
this right over here. And we'll do that
in the same color. So let's say that b
is right over here. F of b is going to be equal to--
we just literally replace the b where you see the
x-- it's going to be equal to the definite integral
between c and b of f of t dt, which is just
another way of saying the area under the
curve between c and b. So this F of b,
capital F of b, is all of this business
right over here. And from that, we
are going to want to subtract capital
F of a, which is just the integral between
c and lowercase a of lowercase f of t dt. So let's say that this
is a right over here. Capital F of a is just
literally the area between c and a under the
curve lowercase f of t. So it's this right over here. It's all of this
business right over here. So if you have this blue
area, which is all of this, and you subtract out
to this magenta area, what are you left with? Well, you're left with this
green area right over here. And how would we represent that? How would we denote that? Well, we could denote that
as the definite integral between a and b of f of t dt. And there you have it. This right over here is the
second fundamental theorem of calculus. It tells us that if f is
continuous on the interval, that this is going to be
equal to the antiderivative, or an antiderivative, of f. And we see right over
here that capital F is the antiderivative of f. So we could view this as
capital F antiderivative-- this is how we defined capital
F-- the antiderivative-- or we didn't define it that way,
but the fundamental theorem of calculus tells
us that capital F is an antiderivative
of lowercase f. So right over here,
this tells you, if you have a definite
integral like this, it's completely equivalent to an
antiderivative of it evaluated at b, and from that, you
subtract it evaluated at a. So normally it looks like this. I've just switched the order. The definite integral
from a to b of f of t dt is equal to an antiderivative
of f, so capital F, evaluated at b, and from that, subtract
the antiderivative evaluated at a. And this is the second part
of the fundamental theorem of calculus, or the
second fundamental theorem of calculus. And it's really the core of
an integral calculus class, because it's how you actually
evaluate definite integrals.

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