# Fundamental theorem of calculus (Part 1) | AP Calculus AB | Khan Academy

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.

## 100 thoughts on “Fundamental theorem of calculus (Part 1) | AP Calculus AB | Khan Academy”

1. fresh1098 says:

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.

2. hippiechickie18 says:

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

3. Curtis McDonald says:

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?

5. Neelav Dutta says:

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

Personal misconception officially gone. Thank you very much.

6. xusedteabagx says:

Eek now I'm just plain confused! =(

7. Seth Aandewiel says:

you son are a master at drawing on a computer

8. neel auporajita says:

Oh God!!!……… I hate Math……….

9. Juliya Pavlovych says:

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

10. Papaconstantopoulos says:

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

11. csanchez505 says:

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.

12. csanchez505 says:

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.

13. Cosmo says:

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!

14. simonjeste says:

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.

15. lucyferrabee says:

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?

16. Kristina Chepak says:

you're a genius

17. Daniel Frisch says:

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

18. David Yang says:

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

19. Arither23 says:

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.

21. Gurjot Kheeva says:

Thanks…so much!

22. Hallie Richie says:

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

23. Omar Allheebi says:

Thank you

24. Alex schwalb says:

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

25. Rushi Patel says:

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

26. Kevin223 says:

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??

27. Nathan Alspaugh says:

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

28. HoiNam Lee says:

Brilliant! This video really helps me to understand the concept

29. Saj Sanilkumar says:

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

Thanks a lot. Helped me immensely.

31. Marc Rover says:

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

32. iDaMelon says:

What are u using to graph it? What program

33. Mario852 says:

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

34. Dick Jones says:

I like soup.

35. ender2999 says:

36. Elias Levinson says:

shooter mcGavin is that you?

37. Andrew Norris says:

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

38. Ilyas Khan says:

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

39. Keanu Oliva says:

THIS IS THE SECOND FUNDAMENTAL THEOREM OF CALCULUS.

40. shashi thapaliya says:

can u suggest any video for leibnitz theorem of derivative????
it would really help me out……

41. shashi thapaliya says:

can u suggest any video for leibnitz theorem of derivative????
it would really help me out……

42. ebscoHOST says:

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

43. Dickson Butts says:

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?

44. Robert Brandywine says:

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.

45. Aimee Coleman says:

I watched this instead of going to my lecture

46. forever Hungry says:

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

47. NW Monk says:

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

48. Whyx says:

Is calculus like this university level in America?

49. Kenneth Wong says:

Thank you for making math so much more interesting! 🙂

50. gill says:

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

51. hg2 says:

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.

52. EDUARDO12348 says:

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

53. George Avetisyan says:

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

54. Ben Lyman says:

Beautiful. A work of art

55. olivejuice says:

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.

56. Modi A says:

can i borrow your brain for two semesters pls

57. Michael Katt says:

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.

58. Andi Sanchez says:

why are you always repeating everything

59. Andi Sanchez says:

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

60. Mazidox Gaming says:

This guy repeats word so much.

61. Learn Maths says:

pretty cool

62. Bongani C Magwaza says:

whats a power of mathematics.

63. Nikhil Leeroy says:

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

You are a lifesaver!

65. EDUARDO12348 says:

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.

66. conner hartman says:

what program is he using?

67. Annette Hoiseter says:

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

68. Liaomiao says:

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

69. Ham24brand says:

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

70. CVRIV ALL DAY says:

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.

71. Quanta Ali Ahmed Jan says:

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….

72. Ashton Beattie says:

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

73. debendra gurung says:

Ok , but what happened to the let bound a

74. Crystal Stone says:

75. JohnVoland says:

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.

76. Logic Λόγος says:

🔊 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.")😆

77. daniel says:

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) ???

78. TheChosenOne says:

79. Blamblaster Sidge III says:

tHE fUNDamenTAL theoReM oF CAlcuLUs

80. Christopher876 says:

Why 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.

81. Emily says:

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

82. George Donald says:

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

83. Hamza B says:

amazing explanation. bravo

84. Halis Karakurt says:

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

85. Isaac Newton says:

Light work

86. Reem B says:

Ughh I love you

87. Bogdan Velickovski says:

Really appreciated the help!!

88. Bender Bending Rodriguez says:

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

89. Gourav Mathematics says:

Nice video
#Mathclassonline

90. DwolfOG says:

Beautiful.

91. parikshit uniyal says:

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

92. Vaibhav Sharma says:

Fundamental theorem is important for AP calculus AB/BC .
Thanks a lot .

93. Angelea Godwin says:

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

94. Milan Šekularac says:

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.

95. Edward Bickerton says:

This guy sounds like Eric Weinstein 🤔

96. Mikaizer Gaming says:

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

97. Michael Shinagawa says:

God bless you 🥰

98. Ford says:

i dont get it

99. Muzakkir says:

I like how he says, THE FUNDAMENTAL THEOREM OF CALCULUS

100. Gary Dreyer says: