18-01F07-L33

18-01F07-L33 - 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 33 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: So again, welcome back. And today's topic is a continuation of what we did last time. We still have a little bit of work and thinking to do concerning polar coordinates. So we're going to talk about polar coordinates. And my first job today is to talk a little bit about area. That's something we didn't mention last time. And since we're all back from Thanksgiving, we can certainly talk about it in terms of a pie. Which is the basic idea for area in polar coordinates. Here's our pie, and here's a slice of the pie. The slice has a piece of arc length on it, which I'm going to call delta theta. And the area of that shaded-in slice, I'm going to call delta A. And let's suppose that the radius is a. Little a. So this is a pie of radius a. That's our picture. Now, it's pretty easy to figure out what the area that slice of pie is. The total area is, of course, pi a ^2. We know that. And to get this fraction, delta A, all we have to do is take the percentage of the arc of the total circumference. That's delta theta / 2 pi. This is the fraction of area -- sorry, fraction of the total circumference, the total length around the rim. And then we multiply that by pi a ^2. And that's giving us the total area. And if you work that out, that's delta A is equal to, the pi's cancel and we have 1/2 a ^2 delta theta. So here's the basic formula. And now what we need to do is to talk about a variable pie here. That would be a pie with a kind of a wavy crust. Which is coming around like this. So r = r (theta). The distance from the center is varying with the place where we are, the angle where we're shooting out. And now I want to subdivide that into little chunks here. Now, the idea for adding up the area, the total area of this piece that's swept out, is to break it up into little slices whose areas are almost easy to calculate. Namely, what we're going to do is to take, and I'm going to label it this way. I'm going to take these little circular arcs, which go. So I'm going to extend past where this goes. And then I'm going to take each circular arc here. So here's a
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18-01F07-L33 - MIT OpenCourseWare http:/ocw.mit.edu 18.01...

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