New calc video! Integrals and the fundamental theorem of calculus

The animation of the Pi Creature's whiplash in the impossibly moving car was an adorable detail.

Sean Bibby

At the moment, I don't plan to. They are certainly worthy, and a full series really ought to give some justification for those processes, but they just got lower prioritization than some other concepts.

3blue1brown

Thanks for the tip, I'll check it out.

3blue1brown

I really like BetterExplained's explanation of definite integrals as basically "better multiplication", or "multiplication of two changing variables". It helped me a lot when dealing with some Physics applications of integrals, as it was much more intuitive than thinking about the area under a curve. Maybe you could mention it at some point!

Are you going to go through integration by parts and by substation? I always liked doing those integrals but I never understood why they worked. For instance why do you differentiate u to get from dx to du, and why does it work?

Royal with Cheese

I thought about it, but I'm just not sure it actually adds anything to the essential understanding of integrals. What do you think? I can't help but feel that most of the time a student has with Riemann integrals is spent just trying to parse the notation, but the only reason we bring up the notation is to illustrate how the idea of "area under a curve" is formalized. My take is that if the same idea can be expressed with different words and more visuals, without all the overhead of notation to parse, it's an efficiency win.

3blue1brown

Great video. Is there a reason why you didn't include the step of constructing the Riemann sum and taking the limit of it to get the integral? It feels like a gap. You practically spelled it out and I don't see the harm in adding a minute or two to put it into symbols.

In answer to your question, yes, I think it's perfectly fair to have a "little review" :)

Karin Rodrigues

I agree, fixing it now...

3blue1brown

Thanks for the catch!

3blue1brown

Hi Josh, thanks for the feedback. You bring up a good point, and I might add a quick sentence to emphasize it.

3blue1brown

In the repeated animation of thinner and thinner slices under the curve, there's a narrow gap at t=6. A small detail, but distracting - every time it played my attention was on that silly gap.

Martin S

at 11:15 there is some faint writing in the background, just above the dA, which probably shouldn't be there

Andreas Blatter

Hey, Grant! Awesome video as usual. I think the only thing for me that made me have to pause and rewind was your explanation around 14:30ish about subtracting the antiderivative evaluated at the lower bound. I think another approach that may make more sense to people is showing how we're evaluating the antiderivative at it's start and end points, and I learned it like you're taking where you end and subtracting where you started to get the distance traveled. I don't think it is as intuitive to say that the reason why FTC part 2 has that subtraction is to ensure that it's zero when integrating from a to a. There's the important distinction that the integral of velocity wrt time is giving you only displacement, rather than position. Also, I'd love to see a mention of indefinite integrals and how those aren't actually tied to areas - from my understanding, they just express antiderivatives, but when given bounds, these become definite and represent areas.

Josh B.

8:23 Interrobang!

PseudsPie