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Calculus: The Washer Method across the x-Axis


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  • Type: Video Tutorial
  • Length: 13:11
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 141 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: Disks and Washers (5 lessons, $8.91)

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
Disks and Washers
The Washer Method Across the x-Axis Page [1 of 3]
All right, let's take a look at another solid formed by a revolution around a particular axis. So first of all, let me describe the region that I want to revolve. So let's take the following curves and look at the region bounded by these things. y = x^2 + 1, y = , x = -2, and x = 2. So these are four different curves that are going to now plot off a particular little piece of two-dimensional land. Now what I want to do is take that region, whatever it is, and I want to revolve it around the x-axis. So I want to think of the x-axis as a hinge, and whatever thing I have, I'm going to just revolve it around. So let's see what it looks like. The first thing I'm going to do is just draw a picture of this. Again, really drawing a picture is the - well I don't want to say the only way to go, because that sounds a little didactic, but it's the only way that I go. I mean, you can do whatever you want.
Anyway, let's see, so this is going to be the regular happy-faced parabola, but it's shifted up one unit. What is the shifting up one unit color? So I'm going to put my parabola right here and make it a little bit more wide-bodied than normal. Well I have to definitely try to get symmetry, because I want this to be really [inaudible]. So there's the parabola thing right there.
Then I have the line, y = . Now this height is 1, right? Because if I plug in zero in for x, I see y = 1. So y = is actually a horizontal line that's going to actually come right half-way in here, just like that. And x = -2 is way out here, and then x = 2 is way over here. And then the line comes in here. And so those four curves actually describe this region right here. Notice that region doesn't actually go all the way to the floor, because I have that y = line down there.
Now, if I take that region - imagine that this whole world was on a region - so I take that region and I rotate it around the x-axis, which is what I'm told I'm supposed to do in order to generate this three-dimensional surface. Then what happens? I take this orange thing, and it revolves around. And you see what it looks like? It actually looks like something that's not going to be a disk like, because actually now this thing will have a hole in it. Can you see the hole? Let me try to draw it for your right now live. It doesn't exactly look like a pulley now. Now, it has this general shape. Do you see, there's a hole, actually, right in the center of it. It's as though you've taken an apple and imagine, now, coring out the center of the apple. If you core out the center of the apple, you're left with a hole. And so what happens now if you start to slice this? Well, you get this. Well, no, we don't get a disk. What I'm seeing now is actually a disk and then a disk cut out. And in fact, if you look at this graph, you see that's exactly what we're seeing here. We're seeing disks here, and then a disk cut out. That's exactly what's going on.
What does that look like? Well, it doesn't really look like just a disk anymore. It sort of looks like, in fact, like CDs. This is an example of sort of like CD stuff, where if you make a slice, what you really see is a disk and then a disk cut out in the middle. Now, these things are called, not surprisingly, washers. And the reason being is because the washers are sort of the ideal example of these things. So you have these disks that are cut out, and when you put them all together like this - now, of course, these washer's are of different size. In fact, you know what this sort of looks like? This looks like exactly what the innards of a pulley look like, because the innards of a pulley not only have this sort of curvy thing so the rope goes through it, but it's got to have a hole in order for the pulley to spin around. So in fact, this is a genuine innard of pulley thing. And the question is what's the volume of that?
So to do that, we can actually just use a technique called integration by washers, but really, it's nothing more than the old idea. All I've got to do now is figure out exactly what the shape is that I'm trying to find the volume of. So let's put these washers together and see what I get. How am I going to slice? Since I'm revolving around the x-axis, I'm going to make a slice perpendicular to the x-axis. And let me put in an arbitrary slice. It's not one of these endpoint slices, but an arbitrary one, so we can examine it. This is a slice of the object, and then there's that little hole here. You get to sort of see the back of the hole there. It's exactly this. That's what I'm seeing.
What do I have to do? I've got to sum up the volume of those washers. So how can we figure that out? We figure that out by just thinking about how we could figure out the volume of that. So the volume would equal - well, I'm going to sum up. Now how am I going to arrange those washers? Am I stacking the washers this way? No, I'm not stacking the washers this way. I'm stacking the washers this way. This is how I' stacking the washers. So I'm going to start stacking way over here, on the left, and stack all the way to here, on the right. So where am I stacking from? I'm stacking from -2 - that's the left-hand most endpoint - and I'm going to stack, stack, stack, stack, stack until I get to +2. So this is going to be, now, the sum. So I'm going to sum up from -2 to +2. What? The volume of this washer.
Now what's the volume of the washer? It's actually pretty easy. It's just the volume of the outer disk minus the volume of the inner disk - what I take away. So I've got to take the area of this circle and subtract off the area of the inside circle. So that's all there is to it, so let's think about how I'd write that down. First of all, what's the thickness of this washer. Since I'm slicing perpendicular to the x-axis, there's a tiny change in which direction? The direction is x, so in fact this is going to be a dx change - tiny change in x - so I'm going to put a dx way down here. And all I've got to do now is figure out the area of this washer - so the face of it.
So first, I've got to find the area of the outside disk, so what do I do? I'm going to scan x going from -2 all the way to 2. So let's pick an arbitrary point. Suppose it's the one I'm drawing right here. Suppose this is at x. So suppose that x is where I made my slice. If I made my slice at x, what's that outer radius? That outer radius, looking at the picture, is what I get if I plug into the parabola. That height represents the radius of this big circle. So I go up to here - this is going up to the parabola - if I'm at a point x, then this length right here is just going to be x^2 + 1. So that is the radius. So the area of this is going to be that squared times . So I have that r squared, and the r is just x^2 + 1, but squared. Let me say that again. This is just r^2. It's the area of the entire large disk, without me cutting this away yet. And the area of the entire large disk is r^2, and so I've got the , and then what's r? If I'm at a point x, this entire thing represents the radius, and that's given by this and I square it.
Now, to remove the inner disk, I have to subtract off the area of the inner disk, and that's r^2 again, but now this r is just going up to this horizontal line and that horizontal line is located at . So in fact, this is just going to be , which is . So that inside cylinder that I'm cutting out just has a radius of . So squared is a fourth.
Well, that's the integral I have to evaluate, and as usual with these questions, once you get the integral, you're home free. The rest of it is actually pretty easy. It's setting up the integral that's hard. So washers - they're nothing more than just disks were I actually subtract disks right from them. Right? A washer is nothing more than a disk with a little disk removed. So use the exact same idea. You just find the area of the big circle, but then subtract out the area of the little circle. That's the technique of washers.
Okay, and so now how would you evaluate this integral? Let me do this for you really, really fast. What I would do if I were me - well, first what I would notice is this is extremely symmetric. It's a happy-faced parabola that's centered around - the axis of symmetry is the y-axis. Since I'm going from -2 to 2, what I do from zero to 2 is actually the exact same thing as what do from -2 to 0. It's the same thing, just reflected. So by perfect symmetry, I can actually just integrate from zero to 2 - just do half of it - but then multiply that answer by 2 to get the other half. That only works if the thing is completely symmetric.
So I'm going to put a 2 in, in order for me to compensate for the fact that I'm doing only half the question. This , though, I can pull out as a multiplicative constant. And now I'm left with (x^2 + 1)^2. So what I do? Well, I'll square that out really fast - x^4 + 2x^2 + 1, and then here. And then you just integrate. I'll do a little bit more of this just for fun. So the integral of x^4 is . The integral of 2x^2[ ]is going to be . The integral of 1 is just x, and the integral of is just .
So that's the indefinite integral, and I evaluate that from zero to 2. So I plug in 2 everywhere. So I have + + 2 - (2). And I subtract off what I get when I plug in zero. And you can see, when I plug in zero everywhere, this just gets to be zero. That's just sort of the nice thing about cutting the integral in half when you can do it. It saves you from actually doing some more evaluation.
So if you plug all that stuff in and figure it out, I think you get . I think that's right. But certainly, this is all correct, and you can just check the arithmetic if you want.
Anyway, that is the volume of this innards of a pulley, and the method was really just to actually sum up washers. So imagine taking a whole bunch of washers - those are two concentric disks, on big disk and on little disk that I removed - and find the volume of all of them and then stack them all together - in this case, from left to right. And the volume of any individual washer is just the thickness, which in this case, is just a tiny change in x - so that's the dx - and then I multiply that by the area of the surface. And to find the area of a surface of a washer, you just find the area of the entire large circle and then subtract of the area of the little key circle. And the area of the large circle, in this case is r^2[ ]where the r is this length right here - and so I square that - minus the area of the little teeny circle, which is r^2, but now, the r in this case is always constant. In fact, notice, for example, in this particular example, the radius of the inside piece is always a half, because I'm at the line, y = . So in fact, it's like I took the pulley innards and I just drilled a hole right through it. There's no variation in the size there. It's just always going to be as the radius. So when I square that, I always get a fourth. So I get for that piece right there. You could imagine a question where, in fact, the radius might change a little bit, and then you plug in that function squared. In this case, it was the constant function squared and I get .
Anyway, more important than thinking about the formulas is just to think about visualizing it and what's going on and figuring out areas of circles. That's all there is to it. Okay, have some fun with washers and core away. See you at the next lecture.

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