18-01F07-L18 - 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 18 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 we're going on to the third unit here. So we're getting started with Unit 3. And this is our intro to integration. It's basically the second half of calculus after differentiation. Today what I'll talk about is what are known as definite integrals. Actually, it looks like, are we missing a bunch of overhead lights? Is there a reason for that? Hmm. Let's see. Ahh. Alright. OK, that's a little brighter now. Alright. So the idea of definite integrals can be presented in a number of ways. But I will be consistent with the rest of the presentation in the course. We're going to start with the geometric point of view. And the geometric point of view is, the problem we want to solve us to find the area under a curve. The other point of view that one can take, and we'll mention that at the end of this lecture, is the idea of a cumulative sum. So keep that in mind that there's a lot going on here. And there are many different interpretations of what the integral is. Now, so let's draw a picture here. I'll start at a place a and end at a place b. And I have some curve here. And what I have in mind is to find this area here. And, of course, in order to do that, I need more information than just where we start and where we end. I also need the bottom and the top. By convention, the bottom is the x axis and the top is the curve that we've specified, which is y = f(x). And we have a notation for this, which is the notation using calculus for this as opposed to some geometric notation. And that's the following expression. It's called an integral, but now it's going to have what are known as limits on it. It will start at a and end at b. And we write in the function f(x) dx. So this is what's known as a definite integral. And it's interpreted geometrically as the area under the curve. The only difference between this collection of symbols and what we had before with indefinite integrals is that before we didn't specify where it started and where it ended. Now, in order to understand what to do with this guy, I'm going to just describe very
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This note was uploaded on 01/18/2012 for the course MATH 18.01 taught by Professor Brubaker during the Fall '08 term at MIT.

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18-01F07-L18 - MIT OpenCourseWare http:/ocw.mit.edu 18.01...

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