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# Arc length intro

AP.CALC:
CHA‑6 (EU)
,
CHA‑6.A (LO)
,
CHA‑6.A.1 (EK)

## Video transcript

we've used definite integral's to find areas what I want to do now is to see if we can use a definite integral to find an arc length what do I mean by that well if I start at this point on the graph of a function and if I were to go to this point right over here not in a straight line we know already how to find the distance in a straight line but instead we want to find the distance along the curve if we were to lay a string along the curve what would be this distance right over here that's what I'm talking about by arc length by arc length and we could think about it is okay well that's going to be from x equals a to x equals B along along along this curve so how could we do it well the one thing that integration integral calculus is teaching us is that when we see something that's changing like this what we could do is we can break it up into infinitely small parts infinitely small parts that we can approximate it with things like lines and rectangles and then we could take the infinite sum of those infinitely small parts so let me break up my arc length into let me break it up into infinitely small sections of arc length so let me call each of those infinitely small sections of my arc length eh-eh-eh-eh I guess I could say a length of differential and arc length of differential I'll call it D s D s now I'm going to draw it much bigger than when at least I conceptualize what a differential is just so that we can see it and what do I mean by breaking it up into these DS's well if that's a d s and then let me do these others in another color that's another change in infinitely small change in my in my arc length another infinitely small change in my arc length if I summed if I summed all of these ds's together I'm going to get the arc length the arc length if I take is going to be the integral of all of these ds's all of these d s is sum together over this interval so we can we can denote it like this but this doesn't help me right now this is in terms of this arc length the differential I we know how to do things in terms of D X's and D wise so let's see if we can re-express this in terms of D X's and DUIs so if we go on a really really small scale once again we can approximate this is this is this is going to be a line we just the way that we approximated area with rectangles at first but if you have an infinite number of infinitely small rectangles you're actually approximating a a non rectangular region the area of a non rectangular region and similarly we're approximating with lines here but they're infinitely small and there's an infinite number of them you are actually finding the length of the curve well just focusing on this as a line for now so this distance right over here I'm just going to try to express it in terms of D X's and dys so this distance right over here that's DX that's it you could view this as an infinitely small change in X and this distance right over here this is a D Y and once again I'm you being loosey-goosey with differentials really to give you conceptual understanding not a rigorous proof but it'll it'll give you a sense of where the formula for arc length is actually coming from so based on this you can see the D s could be expressed as based on the Pythagorean theorem as equal to DX squared plus D Y squared or you could rewrite it as the square root of DX squared plus dy squared so we can rewrite this we could say this is the same thing as the integral of instead of writing D s I'm going to write it as the square root of DX squared DX squared plus dy squared plus D Y squared once again this is straight out of the Pythagorean theorem now this is starting to get interesting effort in terms of DX and dy is but they're getting square there are no radical sign what can I do to simplify this or at least write it in a way that I know how to integrate well I could factor out a DX squared so let me just rewrite it this is going to be the same thing as the integral of the square root I'm going to factor out a DX squared DX squared x times 1 plus dy over DX squared notice this and this is the exact same quantity if I distribute this DX squared I'm going to get this right up here and now I can take the DX squared out of the radical and so this is going to be this is going to be the integral of let me write that in that white color the integral of 1 plus dy DX squared this is interesting because we know what dy DX is this is the derivative of our function dy DX squared and if you take the DX squared out of the radical the square root of DX squared is just going to be DX it's just going to be DX it's just going to be DX now this is really interesting because we know how to find this between two bounds we can now take the definite integral from A to B since now we are integrating a bunch of D X's or we're integrating with respect to X we can say ok x equals a to x equals B let's take the sum of the product of this expression and DX and this essentially this is the formula for arc length the formula for arc length and if this looks complicated in the next video we'll see that it's actually fairly straightforward to apply although sometimes the math gets hairy if you wanted to write it in a slightly different notation you could write this as equal to the integral from A to B x equals a to x equals B of the square root of 1 plus instead of dy DX I could write it as F prime of X F prime of x squared F prime of x squared DX so if you know the function if you know what f of X is take the derivative of it with respect to x squared add it to one take the square root and then multiply and then take the definite integral of that with with respect to X from A to B and we'll do that in the next video
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