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This guy, Grothendieck, is somewhat of a mathematical idol to me. And I just love this quote, don’t you? Too often in math we just dive into showing that a certain fact is true with long series of formulas before stepping back and making sure that it feels reasonable, and preferably obvious, at least on an intuitive level. In this video I want to talk about integrals, and the thing that I want to become “almost obvious” is that they are an inverse of derivatives. Here, we’ll focus just on one example, which is kind of dual to the example of a moving car that I talked about in chapter 2 of the series, introducing derivatives. Then in the next video, we’ll see how the idea generalizes into some other contexts. Imagine you’re sitting in a car, and you can’t see out the window; all you see is the speedometer. At some point, the car starts moving, speeds up, then slows back down to a stop, all over 8 seconds. The question is, is there a nice way to figure out how far you’ve traveled during that time, based only on your view of the speedometer? Or better yet, find a distance function s(t) that tells you how far you’ve traveled after any given amount of time, t, between 0 and 8 seconds. Let’s say you take note of the velocity at each second, and make a plot over time like this... And maybe you find that a nice function to model your velocity over time, in meters per second, is v(t) = t(8-t). You might remember, in chapter 2 of this series, we were looking at the opposite situation, where you know a distance function, s(t), and you want to figure out a velocity function from that. I showed how the derivative of your distance vs. time function gives you a velocity vs. time function, so in our current situation, where all we know is the velocity function, it should make sense that finding a distance vs. time function s(t) comes down to asking what function has a derivative t(8-t). This is often described as finding the anti-derivative of a function. And indeed, that’s what we’ll end up doing, and you could even pause and try that right now. But first, I want to spend the bulk of this video showing how this question is related to finding an area bounded by velocity graph, because that helps to build an intuition for a whole class of what are called “integral problems” in math and science. This question would be much simpler if the car was moving with a constant velocity, right? In that case, you could just multiply the velocity, in meters per second, by the amount of time passed, in seconds, and that gives you the number of meters traveled. Notice that you can visualize that distance as an area, and if visualizing distance as an area seems weird, I’m right there with you. It’s just that on...