Summer of Math Exposition

Thoroughly clear, I loved the visuals. The symmetric approach at the end is very memorable. Would have been interesting to see an application (one of the integrals solved). Motivation: 2/3 Clarity: 3/3 Novelty: 2/3

Its a solid animation and video, but I thought the proof could have been more in depth

Nice visuals. Clear presentation. I miss an example for an application.

Nice job! This video, in my opinion, lacks a bit of your personality. Remind me some 3b1b’s earliest work. But explaination is very nice! Thank you!

While the visuals were good, the video was an overkill for what was going to show

You do a good job showing the intuition behind King’s rule. But as stated, it’s not clear why we would want to use it. It would probably help to actually work through one of those examples at the end, so we can see the power of the tool.

I learned about the “match stick” definition of an integral when I was in high school and I could follow along with this video for that reason. The video creator however commits the cardinal sin. Beginner calculus students (a lot of them) only learned to think mathematically a year or two before. This means when the variables come thick, fast, superfluously, and undefined that is a recipe for losing students. You don’t need to start a function at i. Start it at 0 or 1. It won’t hurt anybody to use numbers. You can also make the end variable a number. Where you really need variables, explain what each one is. I hate to be repetitive about this, but unexplained variables and extra variables just make everything harder.

The visualization at this timestamp of the video is incorrect, for Reimann Integration, we divide the interval such that the first point is $a$ and the last point is $b$. But in the video, when the integral is divided into $8$ bins, the last bin doesn’t exactly land at $b$, this creates confusion for future proof since now, $x_8= a+8\Delta x\neq b$, in fact, $x_8 > b$

Same will hold later $x_{150}= a+150\Delta x\neq b$ Why does this detail matter? because when you are reversing the direction of integral, you are integrating from $x_{150}$ to $a$, and with higher $n$, $x_{150}$ will tend to $b$, now the endpoint of the integral itself changes while taking the limit. Also, the index of summation has a typo, it should be from $0$ to $150$ not $1$ to $150$.

The second proof with symmetry does work and is nicely done!

Good animation. Strong focus on intuitive visuals with necessary hints on the algebra.

Short and sweet video showing King’s rule as a geometric result. The visuals were very smooth and satisfying. Overall a remarkable 5 minute math video

It’s been a while since my last calculus class, so I don’t remember this law exists. So it’s nice I learned something new. When watching the video, I went from “why should I care about this weird formula” to “the formula is obvious and trivial, why did I even need the video”.

I have some suggestions for improvements, or personal preferences:

I prefer when the definition of something precedes the notation, at least in the video format. I know that in math textbook you usually have “Definition 47: X is a thing that does a thing…”, but that’s a different case. In the video you have sentence “In this first case, x sub i, the coordinate of the i-th rectangle, starts at…” At the moment I hear “x sub i” my brain starts to think about what that might be, if it was defined somewhere etc. While my brain is practically frozen, the video continues. You’ve felt the need to add the definition of it into the sentence, so just changing order wouldn’t make the video any longer, and it would be easier to comprehend. I understand that it was already used in the sum previously, but I’ve noticed that only on second viewing of the video, as I just glanced over it first time and there wasn’t a reason to explicitly mention it in the video. For bonus points, there could be animated arrow pointing and jumping from x_1..x_n while you animate changing i, so it’s clear also to visual people. (I get that some people are more comfortable with notation than visuals or descriptions, but making things clearer can’t hurt, especially if point of video is making things clearer.)

Also, the sentence “x_i starts at x = a” is really cursed. I’ve just barely wrapped my head around what x_i is, now there is some x. What is x, what is “starts at”? x = a basically tries to say that x coordinate of something starts at a. Isn’t x_i the x coordinate of the rectangle? Don’t we use x in the x_i formalism so that it’s intuitively obvious we talk about the x coordinate? Isn’t the **x** = a kind of redundant? The whole video could be summed as “we first sum from the left, then from the right”. It’s not that complicated, it just sounds complicated.

As for the last part of the video, I actually like it more than the first half. The promise of the video was that I’ll see the “geometry” of the rule, but the only geometric thing that was somehow visualized was that we first sum from the left, then from the right, followed by bunch of algebra. I felt like it was waste of Manim. On the other hand, your “footnote” at the end actually provides quite nice intuition that can be nicely animated. It’s kind of expected that you’ll get mirrored function, when the first proof contained summing from the right side. That’s pretty much the same thing as flipping the function and summing from the left, you’ll have to just make sure that you are looking at the function at the right interval. This could be easily visualized: Take f(x). Adding minus in front of x flips the function around y-axis. Adding a would move the function so that the “a” end of the original function moves to 0, and adding b ensures that it’s aligned with the “b” end of the original function. You could animate the fliping and sliding of the function and highlight where those ends move.

Overall, I’ve learned something from the video, but feel like it made simple things unnecessarily complicated.