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

44 thoughts on “Integration and the fundamental theorem of calculus | Essence of calculus, chapter 8”

1. Think we found my brains limit

   What my brain approaches as maths gets harder and harder.

2. What a terrific quote. Extends to so many things in STEM research. My profs would love that

3. The classic calculus misconception is that an integral is the area under a curve

4. thank you sir from nepal

5. These videos are awesome. The animations are great and super intuitive, and 3b1b is a great teacher.

6. Nice explaining

7. 太赞了。比我们之前大学学习的都要有趣生动。我要介绍给我的小孩一起来学习！

8. Thank you so much – the only Problem i still have with this idea is exactly the same that Is puzzling you. How can substracting the 2. integral value from the first one include all the infinitissimal small reactangles for the whole graph? Is there a way to visualize why that Is?

9. 15:39 just copied that directly to my notes, nothing to add or remove

10. Thank you very much, I have been doing calculus and I really needed a video like this!

11. @3Blue1Brown I love your videos but I really wish you would you use symbols instead of a smattering of dt dx dy da dT dF. The d's really really confuse me "uhg which d was he talking about again?".

12. 4:03 In 7th grade I thought I was an idiot because I did not understand how a continuous function with infinite points could ever be calculated, since you would need to calculate the distance traveled on infinitely many points which is just impossible. BUT THIS MAKES SENSE!!!!!! NOW EVERYTHING MAKES SENSE!!!!!! I LOVE YOUUUUU

13. 13:40

14. If you think that's weird seeing distance as area, you can analyse the units:

    m/s × s/1 = (m.s)/s
    The "s" cancels
    m

    There you go, velocity by time gives distance

15. Check out this amazing integral of x^-x from 0 to 1: https://www.youtube.com/watch?v=2Fpu9Ylb9_E

16. This math theorem can also be used to explain the universe. When a three dimensional object is projected to a two dimensional-space, some of its information is lost. In reverse, a higher dimension can include much more information of a lower dimensional world.

17. Great video on integration. Thanks for this video

18. not to be a nitpicker, but isn't that the second fundamental theorem of calculus aka The Newton-Leibniz formula?

19. and you are my idol.

20. Thanks for the video ? I am still pondering how v(t) = t(8 – t), could you please elaborate more or am I missing something?

21. Can't explain how much I love you!

22. 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 …

23. at 7:02, not only it a factor in each quantity that we are adding up it also indicates the spacing between each sample step. wow that is at the height of Grant's explanation. Really appreciate it.

24. Watches first 30 seconds.
    So, if it’s not almost obvious, why try and prove it. Got it.
    Watches video about how 1+2+3+4+5+6…….. = -1/12

    Wait, what?

25. Now look at my teachers…i think they really don't know these things and due to such teachers students don't understand maths and science…thats why students hate…thank you very much for this knowledge…i just cant memorise formulae without understanding..i just can't….thank you very much…??????????????

26. To the people who disliked this video….

    Kill yourselves.

27. If you are watching from Smith's class give me a like 😀

28. when you said you were going to make it obvious why integral is the inverse of derivatives i was like: This video is for me!!!!!

29. When I first watched this video, it made no sense. Now I’m coming back after 2 months and it just sort of clicked for me.

30. I'm still not clear what the dx means in a definite integral. Why do we need it if finding the upper bound minus the lower bound provides us with the area already?

31. this is so good i'm literally crying

32. Sorry for stupid question but I'm still a little confused. Can anyone explain what is the relationship between infinitesimal divided parts with the formula of integration ?

33. V^2-u^2=2as anyone?.

34. It's the best explanation ever in my entire life learning calculus!

35. Can you do a video on double and triple integrals? I’d really like a visualisation of that kind of integrals as just computing them can get really abstract (espacially finding curl in a force field for example).

36. I just had a headgasm

37. My math teacher sent us links to these videos. Nice

38. This explanation is absolutely amazing! It is quintessential to understand the simplified proof, before getting into more rigorous proving. I watch the videos before going to lecture, it is a perfect combination with course materials!!

39. Thanks à lot sir and I wish to add also Arabic for more understanding because I have a problem with English… Thanks a lot for the great explanation

40. @11:10 "The derivative of any function, given the area under a graph, is equal to the function for the graph itself."

    This statement seems to suggest that d[f(x)]/dx = f(x), which is incorrect. @10:30 you said 'ds' represents the sliver of change in the AREA under the graph v(t). ds/dt = v(t) suggests that the rate of change in the cumulative area under v(t) is equal to v(t). Notationally, d[A(v(t))]/dt = v(t) where A(…) means 'the area under'

41. You sir, are amazing. thank you so much for explaining integration so clearly. i could not thank you enough

42. One I don't understand is why aren't we multipepling by T/(dt)

43. Nothing but purity.
    I think Grant knows who Anand Srinivas is.
    Anand like grant is one of the greatest. I learnt high school physics from him.

44. Driving without vision on the street is dangerous. At the end he travels -100m on the y-Axis to the bottom of a cliff.

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