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Calculus: Why Shells Can Be Better Than Washers


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  • Type: Video Tutorial
  • Length: 12:05
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 130 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: Shells (3 lessons, $5.94)

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
Why Shells Can Be Better Than Washers Page [1 of 3]
All right. Now I want to take a look at another example where we're actually going to see an application - or a powerful application anyway - of the washer method for finding volumes of solids of revolution. So here's a particular region in the plane that I want us to consider: y = x^4 + 1. So it's bounded by that, x = 1, x = 2 and the x-axis. And I want to take that region, whatever it is, and I want to revolve it around the y-axis. So take whatever it is and just revolve it around the y-axis, spin it this way, and then see what's the solid it sweeps out. And whatever solid it sweeps out, I now want to compute the volume of that solid.
So where I always begin is with a picture. And so let's start there. Again, the picture need not be particularly accurate, as long as it captures the essence of what's going on. So here we go. Let's go ahead and label these things, by the way. So x = 1, that's over here. x =2, that's over here. So x = 1 is this line. x = 2 is this line. So really, all I'm going to care about is what's going on in between those two lines. So I could draw a lot of this picture if I want, but I'm going to try to be a little bit prudent. Also, the x-axis is here, so that's the important thing.
So if we try to plot this - I'm putting a little Cartesian bib here so that I can graph the graph and then throw it away later - part of it anyway. So what's the graph of y = x^4 + 1? Well, x^4 + 1 is just a happy-faced quartic shifted one unit up. And what's a quartic? Well, that's just a parabola. It's a little bit more extreme. It's a little flatter at the beginning and then sharper as you go up. So it basically looks sort of like a parabola. So for thinking purposes, let's just think sort of parabola-ish, but make it a little bit more extreme so that no one gets mad at us. So I'll make it sort of a little bit more gradual here and then more dramatic here. But then I've got to add the 1, so don't forget. So what I'm drawing here now is just the quartic thing, but what I really want to do is graph the quartic thing plus 1. So I've got to take the happy-faced quartic and shift it up 1, so thank you. Let's try it again.
What a great mistake. You know, I think every time you make a mistake, you should have a little bell that goes off, and then that means that you're one thing closer to becoming a mathematician. So I'm a mathematician like 18,000 times over. Anyway, I go one unit up. There we go. And now from here, I'm going to draw a happy-faced quartic, so I'm going to make it a little more dramatic - so a little drama, and then pick it up, a little drama, and then continue. Great.
Now, I'm looking between 1 and 2 - x = 1, x = 2 - and then the x-axis. So the region, I see - even this goes on in this way too, forever - the region that I see that I'm really interested in is this orange region right here. It sort of looks like a trapezoid that someone sat on, right? Because that's not straight. That's a little curvy. It's like someone sat on it and made it a little bit whatever.
Anyway, that's the only piece I need to really look at, so I'll throw the rest of it away, and now I take this and I'm going to revolve this around the y-axis. So imagine taking this whole thing, just rigidly, with a hinge her, picking it up and just moving it over to this side and then all the way around and seeing what it sweeps out. So what it would sweep out would look something like this. So we've got 1. I'm doing this for you live, isn't it exciting? This is -1, this is -2, and then I'm sweeping it out. Of course, if you're doing it live, you show the cheat lines here. You want to make this really pretty. That really is important to me. I want you to be satisfied. And you have my personal guarantee, by the way - my personal guarantee. If you're not satisfied with this particular lecture, just let me know and I'll apologize. My personal guarantee - not worth that much, but there you go. That's actually what happens when you come back and hit the plane again. But of course, this thing is actually really revolving around all the space. And so at this point, it would be sort of a big circle up there, and then down here, you would see a big circle out here. There's another little inside circle here. Oh, and then there'd be actually a little bridge here too.
So what you see here is sort of like a thickened pasta shell, if you want. If you can imagine a pasta shell where you've trimmed away this side that's been revolved around. And then you sort of throw it out. So it's thickened, though. So it's a little thicker than this - actually a lot of thickness to it.
Let's see what method we could use to actually find the volume of this thing. Well we might say, let's try to use the washer method, right? Let's try to stack in a whole bunch of washers. Here they all are. We'd stack them this way. And we would definitely stack them this way. That would be a dx or dy? It's a small change in y, so this would be a dy problem. And we could use washers. Now why is this not a great idea?
Well, what see what happens if you make the slices this way, if we always get the same kind of looking thing. You see, when I slice down here, in this area, I'm going to hit this line for the inside radius, and that line for the outside radius. So that's pretty systematic. I'll see that as I range up around here. But what happens when I go past that? Once I go past this point, now the washers have a different feel. They actually now go from the curve. That's the inside radius out to this line. So they change shape all of a sudden. And once they change shape, it makes it a lot more complicated to actually evaluate. You could use the washer method by first breaking it up into two questions. One question, washers from here to here. Those washers will take on a certain form. And then from here to here, and the washers will take on another form.
So you could use the washer method, but it would involve two separate integrals, two separate setups and so forth. The question is, can we avoid all that? And the answer is, absolutely yes, by putting in shells. And you can almost see, in this picture, how to put the shells in. You can see the shells right there, how to put them in. So what we're going to do is we're going to actually core. So we're going to use this coring idea to core out concentric circles and thus produce this kind of shell-looking object and will sum up those shells.
So let me draw in an arbitrary core for you. If we put in an arbitrary core, it would look like this. And then I core it out again, so I get the second core, and the second core would give me a thing that would look like this. I come down like this. You wouldn't really see this if it were an invisible noodle - invisible shell. You would see it, but that's how it would look. In fact, it actually does look transparent. Isn't that a transparent look to it? It's really neat. It actually looks like it's made out of glass. It's a glass shell.
And so now that's just an arbitrary point, but you can imagine that they're actually all over the place and when they fill up together, it literally looks like this and it actually produces that particular volume. And this is just one particular slice of it. If I cored it out, I would get this particular core to represent that.
Now, so what do I want to do? I want to set up an integral and I want to integrate now from where to where? So the volume - and again, with all these volume questions, the hardest part is setting up the integral. Usually evaluating the integral is not a big deal at all. So let's see if we can set up this integral. So I'm going to integrate.
Now what I'm going to do is I'm going to let this little piece here - that coring process - where is the first place I core. The first place I core is actually right here - in this case, at this point right here, which is 1. So I'm going to start coring at 1, and I'm going to keep coring and keep coring and keep coring, until I get to the very end, which is 2. So I'm going to core from 1 to 2. If think about it this way, in terms of this picture, where's the thinnest - the shell that has the tightest waist? That happens here. The waist is at a radius of 1. And then the largest waist is going to be at 2.
So I'm going to integrate from 1 to 2. And what am I integrating? I'm going to integrate - I'm going to sum up - the volume of one of these pieces. And what's the volume of one of these shells? Well, the volume of one of those shells is easily seen by taking a shell, like this, and cutting it open. And then you immediately see it's just a rectangular box that's very, very thin. And then how can you describe that rectangular box that's very, very thin? The volume of it, once you undo it, is just going to be base times height times width. And in this particular case, what is the base? If this pasta shell had a radius of r, then when I cut it open, the base is just the circumference. And so I see 2r.
And what's the height? If I call that h, then I have h. And what's the thickness? Well, if you look back at this picture, you can see that this thickness is right there. It's a slight movement in which direction? In the x direction. So this is going to be a dx change. And so now what is the base? The base the 2r. Now what's r? Well, r is the radius. That goes from here all the way out to here. And that's x. Suppose I was at an arbitrary point, x. So I'm going to let the x range from 1 all the way to 2. So the x will start at 1, go all the way to 2. And when I let the x scan, no matter where I am at x, x represents the radius. So I have 2x, that's 2 times the radius - so that's this piece - times the height. If I'm at x, how high do I go? Well, notice that no matter where x is, the height is always up to the function. So in fact, it's going to be given by x^4 + 1. You always have to multiply this by the height, which is x^4 + 1. And then I multiply by the thickness, which is a tiny change in x, so that's just dx.
And so there's the integral. So using the shell method, I produced the integral for the volume of this region. And now all I've got to do is integrate that, and this is really pretty easy. The 2 constant I'll pull out. I've got . And what is that? That's 2.
Notice that now the calculus is like the easiest part of the question. Isn't that amazing? Here you are taking Calculus II, and the calculus part is the real trivial part. It's the setting up that's hard. Notice, this is just a whole bunch of really interesting ideas strung together. And in order to solve this question of what's the volume, we actually have to think about how we're going to do that and the method of shells and so forth and setting it up. Once we set it up, this part is literally routine, and of no real excitement.
So you plug in 2 for x and you see . And then you plug in 1, and you get . And if you work all that out and multiply it by 2and you see 24.
So the volume of this thing is 24 units cubed, since it's volume. And we did this by actually coring the object up into shells, figuring out the volume of a particular shell and then summing all those volumes up.
I think tonight, you should eat pasta for dinner. See you at the next lecture.

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