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The volume of a torus The disk $x^{2}+y^{2} \leq a^{2}$ is revolved about the line $x=b(b>a)$ to generate a solid shaped like a doughnut and called a torus. Find its volume. (Hint: $\int_{-a}^{a} \sqrt{a^{2}-y^{2}} d y=$ $\pi a^{2} / 2,$ since it is the area of a semicircle of radius a.)

$2 \pi a^{2} b$

Calculus 2 / BC

Chapter 6

Applications of Definite Integrals

Section 1

Volumes Using Cross-Sections

Applications of Integration

Oregon State University

Harvey Mudd College

Baylor University

Boston College

Lectures

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07:14

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11:17

A torus is generated by ro…

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09:58

Volume of a Torus Repeat E…

03:05

Volume of a Torus(a) S…

08:26

Surface area of a torus Wh…

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A torus is formed by revol…

17:35

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A torus is formed when a c…

19:51

A torus (doughnut) Find th…

So the disc expert plus y squared lesson to a square is revolved about the line X is equal to be or be a strictly larger than A, and doing that will create a solid shape like doughnut, which we call a tourists. And we want to find the volume of this and were given the hint that the integral from negative eight A of the square root of a squared, minus y squared divide is equal to pi times a squared over two. Since it's just the area of a semicircle with radius, eh? So that hint is probably a good place to start. Um, first, what we need to notice is that we are revolving around this vertical line. And so in this chapter, anytime we wanted to find the volume rotating about Burke Alliance, What we want to do is used on the high are, um I won't use a in this case, are you see, um, some outer radius which we call our and we want this in respect. Why? And we swear that. And then we want some inter radius which we call little our prospective wise wo square the And we interviewed this with respect to you. Why? So first let's re bite this function in terms of or in terms of, Why? So that means we could be right. This here as X is equal to plus or minus a squared minus y squared, square rooted. And remember, this here would be, though positive side and on this side would be negative side. And we actually already know just from knowing what the shape of this crap is, what our balance of integration should be, since we're ranging from values of age today. So just go ahead and write that in real quick. And if you weren't sure that we could just look at the rain or or the domain of Oh, so you have the range of these, please. So it should be negative color, negative A to a so first step, but we want to do is figure out what our inner and outer rady I are going to be. So if we start from this moon line here, we're going to go until we hit our first her, which is going to beat me the positive portion of this so we'll have a little r of why should be so since V is on the right, we're gonna put me there and then we're gonna subtract off the positive of that. So negative a squared like that, then we're going to have a capital R. Why is equal to so I'm gonna do the same thing. But we're going to go until we hit our second curve. So just started here. We're gonna go straight through that, which is just the negative of the square root. So once again, we're starting from begin and then we're going to subtract all the negative of that a square minus y squared. But these negatives here cancel each other out. So we would just have B plus square root of a squared minus wife's would. And now, since we know all of that, we can go ahead and plug in our inner and outer radius into this So high negative a two way. So first we have a B plus the square root of a squared minus y squared, squared and then we're going to the truck. Off are inter radius, which was the minus. This we're route of These were honest, twice weird. And all this needs to be swear. And we integrate with respect to why. And now I don't want to bore you all with the algebra here, so already did it earlier. But it should turn out to the pie. Negative A to a of for being square root of a squared minus y square. Why now? Let's go ahead and pull out that four beat So we get or be pie for the pie Negative A to a of the square root of a squared minus y squared do you want? And now we were told that this here from our hint should be pie a square over too. And that's just from our hint. So we get four b. Hi, that was Well, hi. A square over to And then we can simplify that too. Pie a squared B and I still need a two right there. So the volume of this tourists only rotate around. The access acceptable to be is to pi a squared

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