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Calculus: An Introduction to Arc Length


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About this Lesson

  • Type: Video Tutorial
  • Length: 11:33
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 124 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Applications of Integral Calculus (18 lessons, $26.73)
Calculus: Arc Lengths and Functions (2 lessons, $3.96)

Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

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Applications of Integral Calculus
Arc Length
Introduction to Arc Length Page [1 of 3]
If you're given a function in the plane, a natural question - or maybe it's not so natural, but it's certainly a question that you've heard about now for like a thousand years - is what's the area under the curve? And the integral will give that to you, because you can approximate it by rectangles, or something, sum them up, and then integral provides what the answer is - what's that area? Now I want to ask a question that actually is a little spin on that question. Suppose I give you a function, and suppose I don't really care about the area under the curve between two points. But what if I want to really focus on the curve itself? What about the curve itself?
Suppose I have this function here. Suppose that's y = f(x). And I give you two points. So I give you point out here, let's say a. And I give you point out here; let's say b. I want you to go up to the function here and here. Now instead of asking the question what's the area under the curve, what about the curve? How long is that curve? So what if I want to find the length of that curve? Imagine if this were a road, for example, and you were driving on it. If you were driving on it like this, how far have you driven? That's a sort of interesting question and actually seems like a really hard question. How would you answer that? So I want to talk about how to find out a length of a particular function, or as it's referred to in mathematics, the arc length. So that's the length of this arc. This curve is going to be thought of as an arc - the length of the arc.
And how do we do it? Well, the answer is, I don't know, and so the solution is to not do that but instead do an easier question. So in olden days, when I was trying figure out areas under curves, I'd put in a whole bunch of rectangles and found the areas of rectangles. Let's take that strategy of converting a hard problem into an easier problem and see if we can make this thing a go. So let's now, instead of finding the length of this very, very complicated curve, let's just pick some points and connect them with straight lines. And now finding the length of straight lines, that's easy, right? You just take out rulers and you start measuring away. And you can do that. And that will give us an approximation to what the arc length is - finding the arc length.
And then if I actually put in more lines, each of which were sort of shorter, then I would get even a better approximation. And I put in more lines, better approximation. And if I do that sort of in the limit, I would actually converge right to that beautiful curve. So let's see if we can inspire and derive what that formula would look like for arc length by taking this approximation on.
So what I want to do is what? I first of all want to figure out what that length is. So what is that length? Suppose that this point is an arbitrary point, x. If that point is x, then this would be f(x). So this would be the point f, f(x). And now suppose that this point is really, really near by. In fact, that little offset is just a small change in x. And small changes in x - we usually refer to a change in an x as x. So let's suppose that this little teeny thing right in here, that's just a small change in x. So this point would be x plus that small change. So this would be x + x. Then if you run up to the function, then that point would be x + x, f(x) + x. So that's the height of that point right there.
Now, if I want to find the length of that red line, you could measure it, but how would you measure it using mathematics? We'd use the Pythagorean theorem. We'd use the Pythagorean theorem, because I have a little teeny right triangle. And that leg we know, that leg we know and so I can find that leg. You might actually remember the formula for distance between two points in a plane. I don't even bother with it. I just always re-derive it with the Pythagorean theorem.
This little teeny length there, that's the change in x and we call that x. And what's this thing? Well, that's just the change in y. So I could write that, actually, as change in y if I wanted to. In fact, maybe I should do that. Do you think I should do that? Okay, let's do that. So that's a small change in y. You can actually write that down exactly, if you wanted to. It would just be f(x) + x - f(x).
Now, what would this length be? Well, using the Pythagorean theorem, I'd see that that distance there - so that length, let me call it for length - that length would actually equal +. And that's just from the Pythagorean theorem, because this squared plus that squared should equal that squared. So I see that ^2 = (x)^2 + (y)^2. And then I take square roots of both sides, and since it's an actual length, I know it's positive. There you go.
Now that's the length. That's just the length of that little piece. I want to now sum up all those lengths. So I want to sum - so arc length should be a sum. And I'll start here at a - that's where my first little line segment begins - and I go all the way to b. So I go from a to b, and I'm going to sum up that length right there. Each case I sum length - sum length, length, length plus length plus length plus length. And there's the length, so I sum that up. So it sort of looks like this.
Now this is all slightly sort of semi-fantasy math. Now let's see if we can make it work. So think of this really as sum. In fact, maybe I should write that in - sum. Now it's no fantasy at all. This just means I'm summing up all those lengths. Now what does that give me? What I want to do is I want to make those lengths smaller and smaller and smaller. Let me, in fact, factor out this x^2. If I factor out the x^2 - let me do that as a little side calculation. If I take this quantity and factor out the x^2 - I'm going to take this just slightly slower than usual, just to make sure that you and I are happy - if I factor out of this term, I just get a 1, and if I factor out of this term, well it's not there, so it becomes a denominator. So it would be written this way: . Now I'm going to check this, so just hang with me now. Let's see if that really is okay by distributing. x^2 x 1 is x^2. I'm fine. What about x^2 times this thing? Well, imagine distributing, you see they cancel out, and I'm just left with the y^2. So this is sort of peculiar looking, but it's actually completely legal.
Now the square root of a product of two things is just the square roots of the things individually, so I could write it this way. And actually, let's write it like this, - sort of like a slope-looking thing. And the square root of x^2, they actually kill each other, because the x, I'm pretending now, is a positive number. So it's just x - right? They kill each other. It's like saying . Well, that's just 5.
And so then I see, after just a little teeny bit of arithmetic, that this length can be written this way. I didn't do anything except just use some arithmetic. So let's insert that now right here. So in place of this, I'm going to insert this. So if I do that, I'll bring this up here for a second and block the view but at least we can see.
So then this will equal the sum from a to b of this term.^ I'll write them in the other order, though. So I'm going to write , and then I have a x - a times x. So that's the formula if I have little teeny line segments and I want to sum them all up. But I want to put in more segments and more segments, which means that this small change in x would be shrinking down and approaching zero. So what happens when x approaches zero? When x approaches zero, sums become integrals. So in fact, when we actually let things approach - sort of infinitely many lines that are sort of infinitesimally sort of long - what I see is that arc length would equal now an integral from a to b of what?
So let's let the parameters all go to zero. Here I'll see square root 1 plus - and so what does slope become? If I take the limit, , that becomes a derivative. If you take the limit, that's . So that's , and this becomes dx.
So we just actually found - we actually derived - the formula that will give us the length of a path - the arc length of a particular function. If I have the function and it equals y, then the arc length is given by . And in terms of the function f, you could write this just a different way. What's ? It's just f'. So just a bookkeeping thing - I could write the formula this way. I could say it's . So that is the exact same thing as this, just different nomenclature. So it's really no new idea.
And it looks like sort of a weird formula if you just look at it. If I just covered everything up - don't look anywhere else, just look at that formula - that formula looks weird. This is definitely a weird-looking formula, no doubt about it. But it actually can make some sense if you think back to how we got it. We just put in a whole bunch of little line segments. And then once we're trying to find the length of a line segment, that required the Pythagorean theorem, and then all way did was some little bit of algebraic tricks to write the Pythagorean theorem in this fashion, and then we actually see the formula.
So the square root and the square term here is nothing more than the Pythagorean theorem hidden, in disguise, inside this arc length formula. And so up next - in fact, why don't we actually just try to compute the arc length of a particular function and see how we'd actually use this formula in practice in actually finding the length of a path. Neat. I'll see you at the next lecture.

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