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# Fundamental theorem of calculus

The fundamental theorem of calculus shows how, in some sense, integration is the opposite of differentiation. Created by Sal Khan.

Video transcript

Let's say I have
some function f that is continuous on an
interval between a and b. And I have these brackets here,
so it also includes a and b in the interval. So let me graph
this just so we get a sense of what
I'm talking about. So that's my vertical axis. This is my horizontal axis. I'm going to label
my horizontal axis t so we can save x for later. I can still make this
y right over there. And let me graph. This right over here is the
graph of y is equal to f of t. Now our lower endpoint is a,
so that's a right over there. Our upper boundary is b. Let me make that clear. And actually just to show that
we're including that endpoint, let me make them bold
lines, filled in lines. So lower boundary,
a, upper boundary, b. We're just saying
and I've drawn it this way that f is
continuous on that. Now let's define
some new function. Let's define some
new function that's the area under the curve
between a and some point that's in our interval. Let me pick this
right over here, x. So let's define
some new function to capture the area under
the curve between a and x. Well, how do we denote
the area under the curve between two endpoints? Well, we just use our
definite integral. That's our Riemann integral. It's really that right
now before we come up with the conclusion
of this video, it really just
represents the area under the curve
between two endpoints. So this right over
here, we can say is the definite integral
from a to x of f of t dt. Now this right over here is
going to be a function of x-- and let me make
it clear-- where x is in the interval
between a and b. This thing right
over here is going to be another function of x. This value is going to depend
on what x we actually choose. So let's define this
as a function of x. So I'm going to say that this
is equal to uppercase F of x. So all fair and good. Uppercase F of x is a function. If you give me an x value
that's between a and b, it'll tell you the
area under lowercase f of t between a and x. Now the cool part, the
fundamental theorem of calculus. The fundamental
theorem of calculus tells us-- let me
write this down because this is a big deal. Fundamental theorem-- that's
not an abbreviation-- theorem of calculus tells
us that if we were to take the derivative
of our capital F, so the derivative-- let me make
sure I have enough space here. So if I were to take the
derivative of capital F with respect to x, which
is the same thing as taking the derivative of
this with respect to x, which is equal to
the derivative of all of this business--
let me copy this. So copy and then paste,
which is the same thing. I've defined capital
F as this stuff. So if I'm taking the derivative
of the left hand side, it's the same thing as
taking the derivative of the right hand side. The fundamental
theorem of calculus tells us that this is going to
be equal to lowercase f of x. Now why is this a big deal? Why does it get such
an important title as the fundamental
theorem of calculus? Well, it tells us that for
any continuous function f, if I define a
function, that is, the area under the curve
between a and x right over here, that the derivative of that
function is going to be f. So let me make it clear. Every continuous function,
every continuous f, has an antiderivative
capital F of x. That by itself is a cool thing. But the other really
cool thing-- or I guess these are
somewhat related. Remember, coming into
this, all we did, we just viewed the
definite integral as symbolizing as the area under
the curve between two points. That's where that Riemann
definition of integration comes from. But now we see a connection
between that and derivatives. When you're taking
the definite integral, one way of thinking,
especially if you're taking a definite
integral between a lower boundary and an x, one way
to think about it is you're essentially taking
an antiderivative. So we now see a
connection-- and this is why it is the fundamental
theorem of calculus. It connects
differential calculus and integral calculus--
connection between derivatives, or maybe I should say
antiderivatives, derivatives and integration. Which before this video, we
just viewed integration as area under curve. Now we see it has a
connection to derivatives. Well, how would you actually
use the fundamental theorem of calculus? Well, maybe in the context
of a calculus class. And we'll do the intuition
for why this happens or why this is true and maybe
a proof in later videos. But how would you actually
apply this right over here? Well, let's say someone
told you that they want to find the derivative. Let me do this in
a new color just to show this is an example. Let's say someone wanted to
find the derivative with respect to x of the integral
from-- I don't know. I'll pick some
random number here. So pi to x -- I'll put
something crazy here -- cosine squared of t
over the natural log of t minus the
square root of t dt. So they want you take the
derivative with respect to x of this crazy thing. Remember, this thing in the
parentheses is a function of x. Its value, it's going to have
a value that is dependent on x. If you give it a
different x, it's going to have a different value. So what's the derivative
of this with respect to x? Well, the fundamental
theorem of calculus tells us it can be very simple. We essentially-- and you can
even pattern match up here. And we'll get more
intuition of why this is true in future videos. But essentially,
everywhere where you see this right
over here is an f of t. Everywhere you see a
t, replace it with an x and it becomes an f of x. So this is going to be
equal to cosine squared of x over the natural log of
x minus the square root of x. You take the derivative of
the indefinite integral where the upper boundary
is x right over here. It just becomes whatever you
were taking the integral of, that as a function instead of
t, that is now a function x. So it can really simplify
sometimes taking a derivative. And sometimes you'll see on
exams these trick problems where you had this really
hairy thing that you need to take a definite
integral of and then take the derivative,
and you just have to remember the fundamental
theorem of calculus, the thing that ties
it all together, connects derivatives
and integration, that you can just simplify it
by realizing that this is just going to be instead of a
function lowercase f of t, it's going to be
lowercase f of x. Let me make it clear. In this example right over
here, this right over here was lowercase f of t. And now it became
lowercase f of x. This right over here was our a. And notice, it
doesn't matter what the lower boundary
of a actually is. You don't have anything
on the right hand side that is in some
way dependent on a. Anyway, hope you enjoyed that. And in the next few videos,
we'll think about the intuition and do more examples making
use of the fundamental theorem of calculus.