18-01F07-L22

18-01F07-L22 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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Unformatted text preview: 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 MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Transcript – Lecture 22 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: Today, I'm going to continue the idea of setting up integrals. And what we'll deal with is volumes by slices. By slicing. And it's lucky that this is after lunch. Maybe it's after breakfast for some of you, because there's the typical way of introducing this subject is with a food analogy. There's a lot of ways of slicing up food. And we'll give a few more examples than just this one. But, suppose you have, well, suppose you have a loaf of bread here. So here's our loaf of bread, and I hope that looks a little bit like a loaf of bread. It's supposed to be sitting on the kitchen counter ready to be eaten. And in order to figure out how much bread there is there, one way of doing it is to cut it into slices. Now, you probably know that bread is often sliced like this. There are even machines to do it. And with this setup here, I'll draw the slice with a little bit of a more colorful decoration. So here's our red slice of bread. It's coming around like this. And it comes back down behind. So here's our bread slice. And what I'd like to figure out is its volume. So first of all, there's the thickness of the bread. Which is this dimension, the thickness is this dimension dx here. And the only other dimension that I'm going to give, because this is a very qualitative analysis for now, is what I'll call the area. And that's the area on the face of the slice. And so the area of one slice, which I'll denote by delta V, that's a chunk of volume, is approximately the area times the change in x. And in the limit, that's going to be something like this. And maybe the areas of the slices vary. There might be a little hole in the middle of the bread somewhere. Maybe it gets a little small on one side. So it might change as x changes. And the whole volume you get by adding up. So if you like, this is one slice. And this is the sum....
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18-01F07-L22 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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