18-01F07-L31 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Please use the following citation format: David Jerison, 18.01 Single Variable Calculus, Fall 2007 . (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms
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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Transcript – Lecture 31 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK. Now, today we get to move on from integral formulas and methods of integration back to some geometry. And this is more or less going to lead into the kinds of tools you'll be using in multivariable calculus. The first thing that we're going to do today is discuss arc length. Like all of the cumulative sums that we've worked on, this one has a storyline and the picture associated to it, which involves dividing things up. If you have a roadway, if you like, and you have mileage markers along the road, like this, all the way up to, say, sn here, then the length along the road is described by this parameter, s. Which is arc length. And if we look at a graph of this sort of thing, if this is the last point b, and this is the first point a, then you can think in terms of having points above x1, x2, x3, etc. The same as we did with Riemann sums. And then the way that we're going to approximate this is by taking the straight lines between each of these points. As things get smaller and smaller, the straight line is going to be fairly close to the curve. And that's the main idea. So let me just depict one little chunk of this. Which is like this. One straight line, and here's the curved surface there. And the distance along the curved surface is what I'm calling delta s, the change in the length between, so this would be s2 - s1 if I depicted that one. So this would be delta s is, say s. si - si - 1, some increment there. And then I can figure out what the length of the orange segment is. Because the horizontal distance is delta x. And the vertical distance is delta y. And so the formula is that the hypotenuse is delta x ^2 + delta y ^2. Square root. And delta s is approximately that. So what we're saying is that delta s ^2 is approximately this. So this is the hypotenuse. Squared. And it's very close to the length of the curve. And the whole idea of calculus is in the infinitesimal, this is exactly correct. So that's
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18-01F07-L31 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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