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.

The proof is actually pretty simple – but you need to know the definition of integral for that – I don't really remember it right now.

Tip;If it's hard for you to think about F(x) = integral of f(x) , you can think about it as f(x) = derivitive of F(x) , and try lookin at the definition of derivitive.

That's all fine and dandy, but what is the airspeed velocity of an unladen swallow?

Well African or European.

So why not F(x) – F(a)? I can see why it is f(x) though. Maybe I need to go back a few videos?

So dF/dx = f(x), not f(t)?

Personal misconception officially gone. Thank you very much.

Eek now I'm just plain confused! =(

you son are a master at drawing on a computer

Oh God!!!……… I hate Math……….

all i thought about was how the blue, orange ..or peach and yellow look good together.

Lol, it's true, Sal is actually really good at writing with that mouse, something most of us absolutely cannot do

I believe he is using SmoothDraw (which is free), accompanied by a Wacom Tablet. These tools used together enable him to create these stylish diagrams.

Oh yeah, and he's also using Camtasia Studio for the screen casting. This software will run you a couple hundred dollars, but there is a free alternative called "ScreenCastOMatic" that I really like, but you only have a 15 minute recording ceiling.

Great video! My school does not teach this, since of course I am in grade 7 only. I must say, I found this lesson quite unique… You have made this video so easy to understand, and I am 12.. Thank you very much!

Related video had that exact title, "Proof of Fundamental Theorem of Calculus"

/watch?v=pWtt0AvU0KA

I haven't watched it yet, but it's 14 minutes long so probably is what you are looking for.

what if the upper bound of the integral is in terms of two different integrals- i.e. x and t. would this change the derivative?

you're a genius

Sal "Let me write this down, this is a big deal." XD

If the upper and lower bounds were both x… wouldn't it be 0?

You have your first and second theorems confused. This is the Second Fundamental Theorem of Calculus, not the first. Good teaching, just wrong theorem.

My debt to Khan Academy is one without end.

Thanks…so much!

What program is used to make this video? Is it a special software to write with the mouse and then convert it to YouTube?

Thank you

this is a really weird question, but what microphone did you use? I like its tone

idek why people dislike this video! Khan Academy is really help (for me at least)!

when you replace t s with x variables its it because basically, you are finding the definite integral of that function where its F(upper bound which is x in this case) – F(lower bound). And then taking the derivative of the function you get the original function in terms of x and since the integral of the lower bound gives you a number, the derivative of any number is just zero. so the net effect is like replacing the t with x??

EVERYONE! Watch this video at 1.25x regular speed! It's much faster and just as understandable! Good luck!

Brilliant! This video really helps me to understand the concept

Thank You Khan Academy, this video was a HUGE help!

Thanks a lot. Helped me immensely.

Isn't this kind of like saying "If you add 1 to 1 you arrive at 2. But, wait a second. If you subtract 1 from 2, you get, gasp, ONE." Obviously the derivative of an integral is the integral before it's solved. Am I missing something here?

I must be missing something. If you solve a function for it's integral OBVIOUSLY the original function is it's derivative?!?!??!?!!??

I'm so freakin confused.

Example: Integral of X^2 = x^3/3 and the derivative of x^3/3 = x^2

What are u using to graph it? What program

wow he spends 8 minutes on that and he didn't prove shit.

I like soup.

Your computer hand-writing is amazing.

shooter mcGavin is that you?

Thank you, you you guys are making this world a better world

Lets say I have some function f that is continuous, continuous, continuous, on an interval, continuous on an interval.

Jk thanks Sal.

THIS IS THE SECOND FUNDAMENTAL THEOREM OF CALCULUS.

can u suggest any video for leibnitz theorem of derivative????

it would really help me out……

can u suggest any video for leibnitz theorem of derivative????

it would really help me out……

Wow. It finally clicked. Wow. Im on cal 2 and it finally clicked. TY so much! Now off to make this intuitive! :

Is it just me or does this seem pretty underwhelming as far as fundamental theorems go? The fundamental theorems of arithmetic and algebra are incredibly important and groundbreaking. Doesn't this just reiterate the definition of a derivative and integral? Isn't it obvious that the derivative of an antiderivative is just the function?

The geometric explanation of the FTC just sucks. It doesn't click intuitively. As soon as they start talking about the area under the curve you want to say "where the bleep did that come from"? What Wikipedia calls the "physical intuition" is much more meaningful.

Using the physical intuition explanation you start, not with the derivative function but with the anti-derivative function, but you don't call it that, you just call it "some function". Then you look at what you are doing when you calculate the derivative function, which is a whole bunch (infinite number) of divisions — Δy divided by Δx. Now switch your thinking to the derivative function and you can see that undoing this over a definite interval is done by reversing the operation with multiplication. But instead of a whole bunch of multiplications, you are just looking for the TOTAL change over that interval, and that is arrived at by subtracting the value of y at the start of the interval from the value of y at the end of the interval on the function you used to get the derivative function in the first place.

I watched this instead of going to my lecture

i'm a bad student

This explains it well. I think mostly because of the graph. I feel bad for my prof, he's really passionate. However, I can't understand any of them. I guess he expects me to be a genius which I

B is not the high point. Seriously is this what you peddle to kids as math instruction?

Is calculus like this university level in America?

Thank you for making math so much more interesting! 🙂

Does anyone know the notation that is used in the video for the width of the rectangle?

Too bad Sal can't make complete simple sentences instead of having to qualify every word of the sentence and change colors 3 times within the sentence.

Krista King has the best math videos on YouTube.

The fundamental theorem of calculus (FTC) relates differentiation and integration, showing that these two operations are essentially inverses of one another. Before the discovery of this theorem, it was not recognized that these two operations were related. The historical relevance of the (FTC) is not the ability to calculate these operations, but the realization that the two seemingly distinct operations are actually closely related.-Wiki

what if instead of t , there is an x inside of integrant with t : ) lot more complicated for example d/dx integral from 0 – x third root of x+ sin t dt . how about that ? I was trying different methods but couldn't get anything reasonable

Beautiful. A work of art

Absolutely love the contents and the way you cover these materials…but allow me to say that you've got a very distracting voice there.

can i borrow your brain for two semesters pls

I wished I had this available to me when I was learning calculus, college students are so fortunate to have resources like this. There is no excuse for failing calculus these days, and BTW I passed all my calc classes but I had to work really hard at it.

why are you always repeating everything

also you at 6:33 "we'll get more intuition of why this is true in future videos".. anyone know where i can find said videos

This guy repeats word so much.

pretty cool

whats a power of mathematics.

I have my AP calc test tomorrow, and the second multiple choice section on this test is looking brutal

You are a lifesaver!

I learned in https://betterexplained.com/calculus/ that the derivative can be thought as better division, or breaking down the larger pattern into smaller pieces, while the integral is the opposite, which can be thought as better multiplication and finds the larger pattern from the smaller pieces. This relation is the FTOC.

what program is he using?

But you would need to be sure that your f(t) is continous on the intregration-interval though right?

Why does the lower bound not change anything? Be it pi or a or whatever?

how the hell can you write so good with a mouse!? when I do that it's a squiggly.

OMG. Thank you so much! I spent the last two days trying to make sense of this because the book I'm using for class sucks!!! It's so horrible with explanations.

wonderful…I hadn't understand a single word of my instructor, but now it is clear to me what the fundamental theorem is…THANKS KHAN ACADEMY….

wait, isn't that the definition of an antiderivative? F'(x) =f(x)?

Ok , but what happened to the let bound a

Very helpful! Thank you!!

Had to watch this here, the video on khan academy's website doesn't play properly today. Thanks for an alternative place to stream this.

🔊 THE FUNDAMENTAL THEOREM OF

⚡CALCULUS, Calculus, calculus,…⛈🌪🌧🌧

🎵🎶 "Riders on the storm…"🎵🎶

(How it sounds in my head when you say it: echoes with thunder & lightening, then slowly fades out with "the Doors.")😆

6:16 my question is.. does this imply that " x " has to be greater than π,

since in the set up " x " is greater than " a " (i.e. the lower bound of our integral) ???

Read the book…you'll learn more.

tHE fUNDamenTAL theoReM oF CAlcuLUsWhy is it that numbers are not used for like an example? It is so much more difficult to follow when there is not a reference number.

What is the difference between f(t) and f(x)? Are they different functions?

this crap has to be made up lmfao man i'ma fail

amazing explanation. bravo

T H E F U N D A M E N T A L T H E O R E M O F C A L C U L U S

Light work

Ughh I love you

Really appreciated the help!!

I learned more in 8 minutes and 2 seconds than a 50 minute lecture.

Nice video

#Mathclassonline

Beautiful.

I’m sorry but I do not understand may someone pls explain it in even more simplified terms ( I’m in year seven)

Fundamental theorem is important for AP calculus AB/BC .

Thanks a lot .

This guy is awesome and I love Khan academy but I find his repeating of words when writing very distracting.

Edit: after reading the comments the suggestion to increase playback speed to 1.25 really helped alot

The intuitive explanation is rather simple: the rate of change of the area under the curve, from x=a to x, evaluated at x, per small change of x, is in fact the height of the rectangle f(x) delX, i.e value of f(x) itself, at x.

This guy sounds like Eric Weinstein 🤔

Hey man! Tomorrow is my final exam in calculus, please let me borrow your brain?

God bless you 🥰

i dont get it

I like how he says, THE FUNDAMENTAL THEOREM OF CALCULUS

teachers should be required to graduate from khan academy