Main content

### Course: Calculus, all content (2017 edition) > Unit 4

Lesson 14: Fundamental theorem of calculus- The fundamental theorem of calculus and accumulation functions
- Finding derivative with fundamental theorem of calculus
- Finding derivative with fundamental theorem of calculus: x is on both bounds
- Finding derivative with fundamental theorem of calculus
- Proof of fundamental theorem of calculus

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# The fundamental theorem of calculus and accumulation functions

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

## Want to join the conversation?

- can someone help me, i still don't understand this. i'm in yr 11 right now and i have watched this over and over and i still don't get it. if someone could explain and make it simpler. much appreciated .(94 votes)
- Many students find this confusing. Part of the problem is that in almost all our other work we're looking at a graph of the function we're studying. In this case, we're studying a function F(x) but looking at a graph of a different function, f(t). So the first thing I would offer in trying to understand this better is to get a clear picture that this graph does not depict F(x). Instead, F(x) is the area under this graph between point a and point x. As x moves to the right, this area increases, even if f(t) is decreasing. The graph doesn't show F(x) at all; in fact, it doesn't have an x-axis. We're given a function f(t) and asked to think about another function, not displayed, that is the area under the curve, not the value displayed on the curve.

If you get that much firmly in mind, the rest should be easier, but there are a couple of other points of confusion. We're trying to show that if we have a function F(x) that provides the area under the curve f(t), the derivative of F(x) is f(x). Some people have a hard time with this duality of f(t) and f(x). The key to understanding this is to realize that f, by itself, is a function. It can operate on t or x or any other variable or constant within its domain. We set it up initially as f(t) because we're using x for a different purpose (area under the curve) and it would create a logical contradiction for x to be both the independent variable that produces the curve and the independent variable that produces the area under the curve. We use t in place of x for the curve, and sometimes this is called a dummy variable.

The other thing that many people find confusing is that we get the same result no matter what point we choose for a, the starting point of the region we're studying. The reason is that we're trying to get at the rate of change in F(x), and the rate at which F(x) is changing doesn't depend on how much area there is to the left of x.

Ultimately we're trying to prove something that is, in a way, quite obvious: that the rate of change in the area under a curve is given by the value of the curve. When the curve is at y = 3, for example, the area under the curve is expanding at a rate of 3. Makes sense, right?(487 votes)

- When defining Fundamental Theorem of Calculus shouldn't the function of lower limit of the integral be zero? Shouldn't there be '-f(a)'' after f(x) in the right hand side otherwise?(38 votes)
- No, as long as a is not dependent on x.

Assuming that it isn't, you would have d/dx( integral |a to x| f(t)dt)=d/dx (F(x)-F(a))=f(x), because a is a constant and so F(a) is also a constant which, when you take the derivative, becomes 0.(56 votes)

- I was taught in class that this is the second fundamental theorem of calculus, and after searching online I got the same response. Am I missing something here?(28 votes)
- No, you're not missing a thing. :) The Fundamental Theorem of Calculus has two parts. Many mathematicians and textbooks split them into two different theorems, but don't always agree about which half is the First and which is the Second, and then there are all the folks who keep it all as one big theorem. In practice, you can figure out which part people mean by context, so it isn't as confusing as you might fear.(64 votes)

- why is it in terms of f(t), why not f(x)?(25 votes)
- We need to use a variable other than x for the function because we're using x for a point between a and b. Using x for both purposes at the same time would create a logical inconsistency. Bear in mind that we're speaking loosely when we say f(x) is a function. The function is just f, and it produces the same result whether we feed x or t or something else into it. In this context, t is sometimes called a dummy variable because we aren't really interested in t but need something to stand in for x to prevent the logic from breaking down.(63 votes)

- Why lower limit 'a' of integral doesn't show the result differentiating the integral? In other words, why 'a' doesn't influence the calculations?(18 votes)
- That's a good question. In fact, I suspect it gets asked in just about every calculus class. One way to answer is that we're dealing with a derivative of a function that gives the area under the curve. Changing the starting point ("a") would change the area by a constant, and the derivative of a constant is zero. Another way to answer is that in the proof of the fundamental theorem, which is provided in a later video, whatever value we use as the starting point gets cancelled out. We'll compare the value at point x with the value at point x + Δx. Both of those values will include a subtraction for the value at a, so when you subtract one from the other, these values cancel out.(28 votes)

- there are an infinite number of infinitely small time segments between any two moments in time. So I'm wandering how time still flows from a second to the next? I've been studying calculus and just wandering.(5 votes)
- These are mathematical models of reality, not reality itself.

While this is not a well-confirmed idea, and**many**scientists disagree, there are some scientists who think that there is a smallest unit of time, something like the time equivalent of an atom -- you just can't get any briefer a period of time. This unit is the Planck time, which is equal to about 5.39×10⁻⁴⁴ sec.

What is fairly well accepted by scientists is that it is impossible by any means to measure time shorter than 1 Planck time, whether or not there is such a thing as part of a Planck time. So, you could only ever measure whole number increments of a Planck time. But, the dispute amongst scientists is not over the measurement, but over whether there even exists a time period shorter than 1 Planck time.

At present, we don't have the technology to measure time anywhere near as brief as a Planck time (though we have other reasons for knowing that is the limit of measurement). And, if current scientific hypotheses are correct, we will never get close to measuring time that small.

So, while we can create a mathematical model and imagine cutting it into infinitely many pieces just so we can minimize our approximation errors, that doesn't mean that something in the real world can be cut up into infinitely many slices.(32 votes)

- at4:10sal says that every continous function has an anti-derivative.but there are continous functons like sin(x)/x which is undefined only at x=0 .but still it is not possible to find the anti-derivative of this function over any interval which excludes 0....why is that?(9 votes)
- The function sin(x)/x is famous in many ways. But it is not continuous for all x in the real numbers, as pointed out above. At x = 0, the function is undefined. So Sal's statement about continuous functions is correct.

Of course you can take a limit of a seemingly undefined function using the squeeze theorem, l'Hopital's rule, or by using complex variables or other tricks found in multivariable calculus and beyond.

A great summary of sin(x)/x can be found here:

press.princeton.edu/books/maor/chapter_10.pdf(4 votes)

- Um....I'm a bit confused. Isn't derivative the slope of the tangent line at a point in a curve? And isn't integral an area under a curve? So how can we have a derivative of an integral - a "slope of tangent line" of an "area"? Sorry....I'm just really new to calculus....and I just keep trying to visualize everything :-)(8 votes)
- You can interpret the derivative as the "instantaneous rate of change". Thus the derivative of the integral is the "instantaneous rate of change of the area under the curve" as the upper bound on the integral advances.

In theory you could produce a graph to represent the area under the curve as a function of the upper bound, and talk about the slope of that graph, but I've rarely done so.(7 votes)

- is there any proof for the Fundamental Theorem of Calculus ?(7 votes)
- There are many ways to prove that theorem. There is a supposed proof on the site, which in fact is not complete and not completely rigorous, but if you are interested in having a look at it, click on the link below. A good challenge would be to see if you can spot why the proof is not complete, and why it is not fully rigorous. For an additional challenge, see if you can complete the proof rigorously.

https://www.khanacademy.org/math/integral-calculus/indefinite-definite-integrals/fundamental-theorem-of-calculus/v/proof-of-fundamental-theorem-of-calculus(7 votes)

- What is the difference between capital F(x) and small f(x) in this video?

Somebody help !!!(4 votes)- I'm not sure what MrCerias meant by "primitive function". F(x) will generally stand for a (particular) antiderivative of f(x). At2:20, Sal explicitly says what he is taking F(x) to mean.(7 votes)

## 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.