3blue1brownHow to Square a Circle? Geometry of Integral Substitution
I will show you an elegant trick for computing the area of a circle of radius R using integrals. In one dimension, the integral is too tricky. The two dimensional integral of a constant function 1 over the circle is simplified by switching to polar coordinates. But if we are not careful, the integral value changes to 2𝛑R. To fix the mistake, we have to add a local weight factor w(r)=r. This video derives and geometrically explains the integral substitution formula. It shows why it contains the derivative of the substitution function.
Analytics
Comments
6.1
5:48 colours are not very clear
6.9
It's Great but the topic is very common and can add a more talk about jacobian matrix but still a clear cut explanation with no jargon and beautiful flow
4.7
This is a quirky offering, with poor (yet still readable) graphics. It made me think, but I feel that I am somehow being tricked into following something which is obvious. On this basis, it feels like the problem could have been re-expressed in a far clearer way.
5
not a very enthusiastic or exciting presentation, kinda monotone and droning
3.4
This video is a little long. It covers too much. It should not get into the Jacobian.
2.6
Goal Orientation: 2/10
Novelty: 0/10
Thought-Provoking: 0/10
Comprehensibility: 2/10
Technological: 4/10
Overall Average: 1.6/10
3.4
The video doesn't feel very "streamlined", it goes back and forth between single and multivariate integrals.
The topic is quite simple and commonly covered in most calculus 1 classes. This wouldn't be a problem, except for the fact that partial derivatives and double integrals are used in the explanation, and I believe that people who are familiar with multi-variable calculus aren't the target audience for this video.
I also think that the determinant of the Jacobian could have been introduced better, less quickly, and probably don't assume that the a viewer in the target audience knows what the determinant is.
6.3
I wasn't optimistic going into this video, since I feel like it's a topic that has been covered by so many videos before that I doubted there was anything new to add to the conversation. But it did a very nice job of explaining visually and intuitively how the Jacobian factors come up. The most novel thing for me was the decision to present all of the three most familiar avatars of these factors (polar coordinates, u-substitution, multivariable) in one video. It's much more common to silo these off into different calculus courses, and there is some risk in presenting multiple different "levels" of material in the same video like this. But I think the thematic link is strong enough that it pays off. I think this video would work very well, e.g., for a student who has seen all (or most) of these formulas before, but without a visceral enough explanation for them to stick.