Integration and the fundamental theorem of calculus | Chapter 8, Essence of calculus

Intuition for integrals, and why they are inverses of derivatives.
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Timestamps:
0:00 - Car example
8:20 - Areas under graphs
11:18 - Fundamental theorem of calculus
16:20 - Recap
17:45 - Negative area
18:55 - Outro

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Vietnamese: ngvutuan2811

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Various social media stuffs:

[johnhammer]

How cruel it was to learn calculus for years in school and college without knowing the fundamentals. Glad the dark ages of pre internet era is gone. Thanks so much for this videos. You are a gift to mankind.

[RobertMcHalffey]

This channel, The Organic Chemistry Tutor, and Khan Academy are currently giving me the tools to master integral calculus... for free. What a wild time to be alive.

[theflamingsword]

Schools should just adopt this as an intro video and along with some worksheets the lectures can take two weeks off.

[nikoyochum]

Wish these videos existed when I was first learning calculus :P

[prawnydagrate]

I'm just a 12-year-old in 8th grade who likes math and this series so far has fascinated me and I love how these videos won't end without providing a full understanding of what's taught. This series is amazing.

[ThatGuyDownInThe]

being in a car and only looking at the speedometer sounds incredibly dangerous

[shubhamshinde]

Best 10 days in the life of Youtube!!!

[FacultyofKhan]

Friday ... check

Little or no work to do ... check

3blue1brown uploads a video ... check

Today is a good day!

[metrictensor]

I've been watching you for ages and JUST realised that this "group of pi monsters" is made up of 3 Blue and 1 Brown pi monsters....

[FacultyofKhan]

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- Faculty of Khan

[bariumselenided]

I’m so lucky that my calc professor seems to have the same love you do for teaching why things work the way they do, not just teaching how to do problems. His classes are always as engaging to me as your videos. I honestly can’t wait for calc 2 in fall with him.

[FrostyAUT]

School: You gotta learn this.
Me: Why?
School: You gotta learn this.
Me: But what is it good for?
School: You gotta learn this.


A few years later in university ...


Me: Geee, I wonder how I can calculate the area under that curve so I can get the consumer surplus from non-linear demand and supply functions ...

[joseleperez]

Hey, I see lots of comments wishing they had this in their school days and, know what? I'm in my school I'm taking a lot of advantage of this series, THANK YOU!!!

[skatelife]

I'm currently studying mechanical engineering and working as a tutor. I have to say, this series is absolutely amazing! I can only imagine how easier these concepts would have been to grasp if it were to watch your series when I first started learning calculus. I might just incorporate some of your notions in my tutoring sessions, if you don't mind!

[Caspar__]

I have watched this video 4 years ago, while I was in 9th grade and now I am studing math at university and watching it again. What a journey

[kjekelle]

0:00 intro
0:55 distance from velocity
2:27 area under a graph
4:08 approximating and refining
6:29 writing an expression: the integral of v(t)
8:30 how does this help?
9:34 the area as derivative
11:17 the antiderivative
14:57 the fundamental theorem of calculus
16:18 recap
17:46 signed area
18:55 outro & sponsor

[pableraspfgpfg]

Great explanation. I am an aeronautic engineer that loves calculus but until now, it had been quite difficult for me to deeply understand why the integral is the "antiderivative" of a function.
Thank you for your great work!

[chrismain]

I love that the gradient of the area rectangles animated the idea of smoothness. When there are 8 large rectangles, each rectangle's color is noticeably different from the color of the adjacent rectangle. As the rectangles get smaller (approaching the area under the curve) the colors approach a smooth gradient as well.

It seems that beautiful and subtle tricks are your forté, and this one is no exception. Thank you for your very helpful videos!

[gregofuente]

Grant is the Richard Feynman of YouTube

[yoavmatia]

this is THE MOST succinct explanation i have ever heard for integrals - as a teacher myself, I tip my hat to you, very well done!