![]() Infinitely small change in x, and this distance right Here, I'm just going to try express in terms of dx's and dy's. Similarly, we're approximating with lines with the infinitely smallĪnd there's infinite number of them, you are actuallyįinding the length of the curve. Rectangles, we're actually approximating a non-rectangular region. ![]() But if we have an infinite number of infinite small We just the way that we approximated area with rectangles at first. If we go on a really, really small scale, once again, we can approximate. Let's see if we can re-express this in terms of dx's and dy's. Together over this integral so we can denote it like this. Is going to be the integral of all of these ds's sum If I summed all of these ds's together, I'm going to get the arc length. Another infinitely smallĬhange in my arc length. Others in other colors, that's another infinitely It up into these ds's? Well, if that's the ds, and then let me do the That when I at least I conceptualize what a differential is, just so that we could see it. Guess I could say a length to differential, an arc length Infinitely small sections of my arc length a, I ![]() Let me break it up into infinitely small sections of arc length. Rectangles, and then we could take the infinite sum of That we can approximately with things like lines and ![]() This, what we could do is we can break it up into Integration, integral calculus is teaching us is that when we see something that's changing like So how can we do it? Well, the one thing that Is okay, that's going to be from x equals a to xĮquals b along this curve. If we lay a string along the curve, what would be theĭistance right over here? That's what I'm talkingĪbout by arc length. Line but instead we want to find the distance along the curve. Know already how to find the distance in the straight This point right over here, not a straight line, we On the graph of a function, and if I were to go at What do I mean by that? Well, if I start at this point What I want to do now is to see if we can use the definitive role We've used the definite integral to find areas. If that is true, it seems that, as long as you want to measure some property of the curve, if you can express it in dx, you're good to go. It seems to me the only way that the Fundamental theorem of calculus holds. The area of that function represent the arc length. Put in other words, the new function that is created is actually a function where we evaluate the area from. Or can it vary? Should we read the Riemann sum as a sum of f(x)dx, or can we read it as the sum of f(x), multiplied by dx?ģ) Even though people in the comments say this is not about area anymore, I like to still see it that way: Sal seems to transform this function from something that is expressed in terms of arc length into a different function that is expressed as area. It is in fact lim dx->0 dx.Ģ) dx represents a fixed quantity, as in: the width of the rectangles are constant for the integral. It represents the infinitely small width of the rectangle. Can someone confirm the following things?ġ) In the integral notation ∫ f(x) dx, dx is not just a notation, but an actual quantity that gets multiplied while summing the Riemann Sum. Together with the Q/A in this forum I think I've been able to tie the pieces together a bit. I think Sal is making a huge jump here, skipping over things we should have learned.
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