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Calculus: Finding Volume with Cross-Section Slices


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About this Lesson

  • Type: Video Tutorial
  • Length: 9:59
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 107 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: Finding Volumes Using Cross-Sections (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.

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Applications of Integral Calculus
Finding Volumes Using Cross-Sections
Finding Volumes Using Cross-Sectional Slices Page [1 of 2]
How often have you gone into a delicatessen and you want some cheese. And they say, "Okay, how much do you want?" And you say, "Well, I want so many pounds or so many ounces," and so forth. What they're really asking you is what is the volume of cheese that you want. So this raises the question, how do you figure out what the volume of cheese is? And, of course, if you think about how cheese is served, it's served in slices. You get a big hunk of cheese, like this. So here's a big hunk of cheese - Colby. And then, of course, you don't eat it like this. It's not like an apple. Instead, what you do is you slice it. So if you want to figure out the volume - in fact, imagine a delicatessen where you go in and they say, "Okay, how much cheese do you want?" And you say, "four ounces." And they go, "No ho much volume of cheese do you want?" Well, now that's a question that requires you to think about what the slices are. So if you want to figure out the volume, for example, of this, how would you do that?
Well, one way of doing that is just to cut up the slices and see what the slices look like. So if you cut up the slices, for example - let's try to do this right now. Now imagine that the slice is very thin and your butcher is very strong. Now even though this shape is really sort of a peculiar shape - it's curvy and complicated - notice that this piece itself is actually not that complicated. In fact, this piece looks like just actually a little sort of rectangular box kind of thing. Finding the volume of this is actually pretty easy, right? It's just base times width times height. And if you put them all together, you see, what would you get? Well, you'd get the cheese.
For example, suppose I cut away a little piece right in here - so I cut this away. We'll just move this to the side for a second, or let it fall. If I cut this little slice right here, I get another slice, which although the dimensions are different, you'll notice it's the same type of formula I can use to find the volume. Right? In fact, it's just a little, teeny, rectangular solid.
So the point is that if you want to find the volume of something, all you've got to do is find, in some sense, just the area of a cross-section, and then just stack them all together. If you stack these all together, you get the whole thing. Now, of course there is the issue of thickness, but that thickness is so tiny, we can just call it a tiny, tiny change in the width. So when you put these together, if I could just figure out the area of this, which is pretty easy in this case, and then find the area of all the other slices - find the area of that, find the area of that face, find the area of that face. Then when I string them all together, all I've got to do is take that area and then multiply by the height.
Now, what happens if I cut the other way? If I cut the other way, then the cross-sectional pieces would take on a different shape, but the same principle follows. If I just take these slices and put them together and make them really, really thin - so thin that they look almost like they're just two-dimensional - like they sort of just rest on the plane. In fact, they barely stand up. Like this one doesn't even want to stand up. Look at that! It's like dominos. It's domino cheese.
Now look, you see, imagine these are really, really, really thin - much, much thinner than I have actually cut them here - if they were so, like, wafer thin, then all you would have, if you looked at it from above, is just this shape. And you found the area of that, and you found the area of that for every single possible slice you could make - really, really thin slices - and you added up all that area, what would you get? You would get the whole thing. So one way to compute volume to actually just figure out the area of a cross-section and then multiply through by how thick it is. That's a way of figuring out volume. And in fact, this fundamental idea of slicing is a very important idea in trying to figure out the volumes of more exotic shapes.
So moving from cutting cheese to looking at cutting up actual mathematical shapes, let me just sort of notice that if you want to figure out the volume of something - so suppose you have some sort of region. It looks like this, let's say - something really wiggly. I'm going to draw a wiggly region right now for you, live. I know you're saying, "Boy, Ed, how can you draw a wiggly region live?" Well, watch and you'll see. This is a wiggly region. It's like a lima bean's side, but then it goes all the way through. So it's sort of like a cylindrical lima bean. It's three-dimensional. I think I'll try to put in some shading here. In fact, that's not bad, is it?
Suppose you wanted to find the volume of this. If you wanted to find the volume of this, what would I do? Well, one thing I can do is to - if I pretend that there are axes here, for example. Let's put an axis right over here. Suppose that this thing starts at a point a and ends right here, at a point b. So it's right in between those two things. If I imagine putting all these slices in - if put in a really, really, really thin slice - let me show you one thin slice. So I put it in one thin slice, it would look like this. So there's one thin slice and there's a little piece hidden in the back there. That's just one little thin slice. Then all I've got to do is figure out the volume of that and sum over the whole region. So I would sum - that's an integral - from a to b, and what do I sum? Well, I'll just figure out the area of this face. So I'll put down the area of the face, and then the thickness. And the thickness is a tiny change, in this case, in the x direction. So that's dx. So that tiny change in the x direction is dx. So in this picture, for example, this here - this length right there - that's dx. And then this area here is the area. So that's the area there. So I take the area multiplied by the thickness and sum it all up. And when I sum it all up, I get the exact volume. So integrating area gives volume.
And this also works, even if you're integrating with respect to the y for it. Let me show you what a picture of that might look like. Suppose you have another shape. Okay, now I'm going to do another shape for you live. I don't want it to be a lima bean shape, but those are the only ones I sort of know. I'll do an amoeba shape. You'll be really impressed by this one. Look at that. And then I'll just show you what the contour there is so you can sort of see that it's all there, which is difficult to see. Not bad.
So this is this solid object. And suppose you want to actually find the volume of that. Well, what you do is, since this is an object that's sort of standing up on its bottom, I'm going to slice this way now. And if I make a level slice, let me show you what it looks like. This is actually a level slice of this surface, right now, coming to you live. That's the top of it, and it has a little thickness as I'm slicing it out - like a piece of bologna, but much more interestingly shaped. If you have bologna like this, definitely don't eat it. You know, headcheese is just a bunch of meat by-products that are stuck together with jell-o. You can make that any shape you want. So this could be headcheese. So you take a slice of headcheese, if you want to find the volume of this, what do you do?
Well, suppose this point here is like c and this point way up here - this height here - is like d. Then the volume would be - I would sum up these slices. I'd sum up the volume of these slices. So I'd sum up from c to d, what? Well, the volume of that is just the area of the top times the thickness. And that thickness now is a tiny change in the y direction.
And so what I would see is the area of a surface - the surface area - dy. And so now, it would be this kind of picture, where in fact the slicing is going in this direction. So I'd have a tiny change in y times the surface area, so different slicing than before.
Anyway, in either of these two methods, we're allowed to figure out what exactly a volume is by just first figuring out an area and then summing up all those possible areas as I migrate from the lowest point to the highest point. And these solids, now - if you have a solid that's built in this way - we can actually now find the volume of that solid. And so up next, we'll take a look at very specific examples, where we can actually see how to figure out what the volume is just by slicing, just like you would slice cheese. All right, so we'll do some more calculus cutting of cheese up in the next lecture. I'll see you there.

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