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Calculus: The Washer Method across the y-Axis


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About this Lesson

  • Type: Video Tutorial
  • Length: 13:12
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 142 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 y-Axis Page [1 of 3]
Well here we are at the Grand Prix, or something, and you can see the cars just zooming by right behind me there. And it's really exciting to watch. And of course, the question that you really think about is not will one of the cars crash and do five flips and then go into flames and then watch the driver scurry out. But the question really is, those cars are going around in a circle in a circle in a circle, and you can see in the back there, that track - they have to elevate the track a little teeny bit to make sure the cars zoom around. You've got the car zooming around really at sort of an angle. And so of course, the question you want to know is, what is the volume - or how much cement - is needed to actually make that thing. Isn't that what you think when you watch these car races? I do. I don't, but let's pretend that we do think about that, because then, it segues nicely into what I want to really talk about, which is the following question. Let's compute the volume of the amount of cement or concrete, or whatever material you care to use, in order to make the racetrack.
So here's some information about the racetrack. We start off on the plane with y = x^2 - 4, and we look at that curve, meaning the x-axis, and then x = 2, x = 3. So look at those three lines and this curve and that actually is going to form a particular region - piece of land in the plane. And then I want to talk that region and revolve it now around the y-axis. So imagine now the y-axis is one big huge hinge. It takes everything here and just slides it over, goes underneath and comes back up. That's going to now roll out, revolve out, a particular surface, and that surface is actually is our racetrack.
So let's try to first graph this. That's the first thing I tend to do. So what's this look like? So y = x^2 - 4. So that's a happy-faced parabola, but it's been shifted down four units, one, two, three, four, so it's looking like this. So that sort of looks like this. So it's a happy-faced parabola shifted down four, doing this live, right now. In fact, let me put this here, just take a little dribble thing. Like you know if you're trying to feed a baby - have you ever fed a baby? You put a little napkin down so they don't get all over your clothes. Okay, so now if we do it, go down to 4 and I'm going to zoom up. Yes, and then this side - now this is the hard part - you've got to make this perfectly symmetric, hopefully. I'm going to cheat, of course. Cheating is completely allowed as long as there's no honor code while you lecture.
Okay, there it is. There's the parabola y = x^2 - 4. Then we have the x-axis. So let's draw in the x-axis. Of course, it's already there, but I'll just emphasize it. Then we have the line x = 2. Now that's going to be a vertical line two units over. But where is that going to be on the parabola? Well, let's plug in x = 2 and see which point on the parabola we're at. If I plug in 2, 2^2 is 4, minus 4 is zero. So 2 is right here. So that's x = 2. And then x = 3 is one more unit over, which is, let's say, right about here. So that's 3 and that's, again, a vertical line. Parabola keeps going, patch that through, goes down like this. And so what we see is there is a little teeny triangularish looking region. It's like a triangle that someone sort of sat on, because this is not straight. This actually is a little bowed, right? Because it's a parabola. So that's the region bounded by those four objects.
And now what I want to do is take that region - by the way, notice that this region here is not even necessary anymore, because it's just this region here. So luckily, I treat it like a baby, so I can throw it away. And now if I take this and revolve it around the y-axis - now remember how this works. Think of it, literally, like a hinge. So whatever's here, I'm going to just pick up and just move over to this side and then all the way around. And so what happens to this orange region? That orange region sort of comes out and goes over and then goes under again. And so, in fact, what does that orange region look like? Well, that orange region would come out just like this, sort of have a reflection on this side now. And what would happen to this point? That point would spin around, and so I'd see a thing looking like this on the top. And then on the bottom, what I would see is this inner thing, and then this outer thing.
So it sort of now looks like a really severe racetrack, the way I've drawn it, because the idea is that the cars are supposed to sort of lie on the inside track there. You see how that goes? So it's very severe. So they're going really, really fast. Anyway, you get the idea. Let's move on from the car metaphor.
Now the question is, what is the volume of that surface of revolution? So how can we think about it? We need to look for an easy way of slicing this thing, and notice that if I slice with respect to the y-axis - so slice up and down - what will every particular slice look like? If I slice at an arbitrary point y - let's slice right at height of y - then notice what I get is something that looks like this. This will be a slice of the racetrack. What goes around, comes around. That's what they say?
Who, by the way, is they? Did you ever hear people say, "Oh, that's what they say"? Don't you ever wonder who they is? There's like five people out there that are being really famous for their expressions and no one knows their names. That's what they say. Well, who's they? But I digress.
Oo! That was not quite right. Well, luckily, you probably can't even see that. Okay, there you go. Maybe that's not that pretty. Let me try to just shade the very top of it so you can see the top of this. It is, in fact, a very nice washer. That's not bad. I think you can see it.
So now you have this nice washer-shaped thing, and in fact, every slice is going to be a washer. Now the inside of the washer is going to fluctuate a little bit. Right? In fact, the very, very top is a washer that's just all insides, so there's nothing left - just a little rim on top. So that inside really is sort of pulled way out. And then here, the inside is sort of pulled way in. So now we're stacking washers in this direction. So now I'm going to sum up the volume of these washers. So I'm going to sum them up and they're going to be like this.
So let's see how we're gong to sum that up. So the volume I'm going to sum up. Now from where to where? Well, from the most bottom to the most top. And so what's the most bottom? Well, the most bottom is, of course, just the floor. It's the x-axis. So that's going to be y = 0. And how high do I go? That's actually a little tricky. Now you might say the height I go is 3 or something, because that's the biggest number here. Well, we've got to think about this now. The highest I go is this point right here. And where is that? That's actually the height when I am at x = 3. So what I'd actually have to do is I have to actually evaluate this parabola when x = 3 and see what that height is. I'm stacking this way, so I've got to find that height value. So when I plug in x = 3 into here, I see 9 - 4 is 5. So in fact, this height - this maximum height - is y = 5 - height of 5.
I'm going to now sum these things up, starting with zero and going all the way up to 5. And what am I summing up? I'm going to sum up the volume of these individuals washers. And what's an individual washer? What's the thickness? I'm stacking the washers this way, so the thickness is actually going to be a tiny change in the y direction. It's an up and down change. So it's a tiny change in y which we call dy. This is going to be a dy integral. So everything should be in terms of y. Let's make sure everything is in terms of y. Now, what do I have to do? I have to figure out the outer radius and find the area of the whole big disk and then subtract off the area of the little teeny disk that I cut away.
So, if I'm at arbitrary point y, what's the outer radius? Well, the outer radius is the radius all the way out to this line right here. And that length right there is just this length right here. It's always constant, no matter where I am. That outer radius doesn't change. It's like a cylinder where I burrowed out some of the stuff inside. So that outer radius is always this length, and what is this? This is zero, and then I go all the way out to here, and that was given to be 3. And so, in fact, that outer radius is just 3. And so what do I see? I see, therefore, the area of that is going to be r^2^^^^. So that's time r, which is 3^2, which is 9.
So the area of the entire disk is just 9. But now I've got to drill out that hole. And if I drill out that hole, what do I see? Well, what's the radius of that hole? Well, now I just go up to the hole part. And if I go up to the hole part - if I'm at y - how will I figure out what this length is in terms of y. Well, I've got now to figure out what x equals when I'm at y. So I'm going to actually take this thing and solve it for x. Now if I solve that for x - I'm going to do that for you really fast here as a side calculation. Why am I solving for x, by the way? Because I want to know, at this height, how far over is this? And that's an x direction. So I've got to figure out what do I need here so that when I plug it into the parabola, I get y. So to figure that out, I've got to solve this for x. So I bring the 4 over.
Technically, I need plus or minus the square root. But since I'm computing a radius, and that's always positive, I can just take the positive square root and I just see + and that equals x. And so that's the radius of this inside disk. So the area that I have to subtract away is equal to times this thing squared. But notice that thing squared lifts the radical. So, in fact, it's just going to be y + 4. That's , the radius, squared. So that's the integral I have to evaluate.
And as always, you can see the integral part is actually really the easiest part to figure out - namely to solve it. The hard part is always to actually set up the integral. But if we now evaluate this really fast, what do we get? The 's I'm going to pull out. It's a constant. The integral of 9 with respect to y is not 9x, but in fact, 9y. And then I have minus, and then I've got this y here, and so the integral of that is going to be . And then I've got another negative. Don't forget to distribute that negativity. And I have a -4, and so the integral of that is -4y. And then I evaluate this from zero to 5. So when I plug in 5, I see x 9 x 5, which is 45, minus minus 4 x 5, which is 20. And then subtract from that what I get when I plug in zero, which is just all zero. So in fact, all you have to do is to plug in the 5, and if I do this carefully, I got , and hopefully that's the right answer. I think that's correct.
Anyway, the important thing is to see how to set this thing up and then carefully analyze the washer situation and realize that, in fact, I'm stacking washers now with respect to y. Now, the question that I want to leave you with is the following.
This little method of solving for x, that seemed a little bit tricky. You had to solve for x in terms of y and then plug in. It was a little, little tricky. Is there another way of looking at this question and slice some other way? I'll leave that as sort of a cliffhanger mystery, but for now, let's just appreciate the power of the washer. I'll see you at the next lecture.^

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