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We've been doing a lot of rotating around the x-axis, so

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let's start rotating around the y-axis and see what we can do.

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Or at least attempt to.

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Let's me draw my axes.

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That's y-axis.

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That's my x-axis.

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Well let's just do it with an example, but we'll call it f

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of x too because it'll be generalizable.

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Let's just draw y equals x squared.

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Let me just draw the positive because we're going to rotate

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it around the y-axis and it's symmetric anyway, so that's

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y equals x squared.

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This is y-axis.

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This is x-axis.

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Actually, no I'm going to keep it general, then we'll actually

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solve it particularly.

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So we'll call this f of x, but clearly this is

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y equals x squared.

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This is f of x.

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And we know how to take the volume if I were to rotate

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this around the x-axis.

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But what if I wanted to say-- I guess we could call it the area

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between 0 and-- I'm trying to determine how general to be.

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Well let's just say between 0 and 1.

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I think the boundaries might make sense to you.

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Roughly this area, and I'm going to rotate it

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around the y-axis now.

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So what's that final figure going to look like?

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The base of it-- let me see how well I can draw it.

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Nope that's not what I wanted to do.

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The base is going to look something like a

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cylinder like that.

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And then the top of it is also going to be-- no, that's

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not what I wanted to do.

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Let me draw the side lines.

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So it's going to look something like that.

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And then the top of it looks something like that.

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But it's not just going to be cylinder, right?

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If I was doing this entire block it would be a cylinder.

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But the inside of it is going to be kind of hollowed out.

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Let me see how effective I am at drawing that.

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I'll do it in a different color.

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So the inside is going to be hollowed out.

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I don't know if that makes sense to you that.

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It's kind of like on the inside it'll look like a bowl.

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On the outside it'll look like a cylinder or a can.

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Hopefully that makes sense.

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You take this and you rotate this around.

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And the curve that specifies the inside would be y

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is equal to x squared.

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It would rotate all the way around.

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I think that makes sense.

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The drawing is the hardest part.

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So how do we do it?

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Well even the shape might give you an idea.

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We can't use that disk method, what we were doing before when

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we were rotating the x-axis, that was the disk method,

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because we were essentially imagining each of these

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particular disks and then summing them up.

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Now we're going to do something called the shell method.

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So what's the shell method?

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Instead of taking a bunch of disks and figuring out their

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combined volumes, we're going to take a bunch of shells.

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So what's a shell?

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So imagine a rectangle right here.

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Hope you can see it right there.

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Let's say it's at the point x,1.

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What's its height going to be?

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Its height going to be f of x,1.

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That's its height.

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Now imagine taking that sliver and rotating

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it around the y-axis.

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What's it going to look like?

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Well, it's going to look like a shell, it's going to look like

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a cylinder, just like the outside of a cylinder.

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It's going to look not too different then that but I want

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to draw it well because intuition is the most important

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thing that, not getting the problem right.

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Let me see if I can draw this respectably.

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And then we're going to have the bottom of the shell, it'll

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look something like that.

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Let me finish these lines up.

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I think you get the point.

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OK.

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So it's going to look like a shell like that.

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The outside of the shell is going to be solid.

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And it'll have some width, but the inside is hollow.

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Let me do a different color.

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Maybe a darker color to show that that's the inside.

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You know it's like a ring essentially.

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And so what's the height of this ring?

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The height is going to be f of x,1.

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So let me do a brighter color so you know what I'm saying.

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The height of this ring is f of x,1.

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f of x evaluated at that arbitrary point we picked up.

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What is going to be the surface area of this ring?

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You know, this outside.

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Well let's think about it.

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It'll be the circumference of this ring times it's height.

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So what's the circumference of this ring?

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Let's go back to our basic geometry.

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Circumference is equal to 2 pi times the radius.

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So if we know the radius of it, we know the circumference.

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Well what's the radius?

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Well the radius is how far we went from the axis of

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rotation to that point.

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So that's the radius.

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So in our particular example the radius is x,1.

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It's that x point that we're evaluating it at.

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So circumference is going to be equal to 2 pi times that point

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that we're evaluating at.

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And so the surface area-- this magenta thing that I filled

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in-- that's going to be equal to the circumference times this

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height, which we already said is f of x,1.

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Let's call it area surface.

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Surface area is equal to circumference times height,

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which is equal to 2 pi x,1 times f of x,1.

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We figured out the surface are of this.

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Now how do we figure out the volume?

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Well what's the width of it?

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How thick is this ring?

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What's this thickness right here?

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It's a very small thickness.

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But we took this sliver, and this sliver as we learned in

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previous calculus, the width of this little rectangle is dx.

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And you know when we take the integral, it's going to get

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infinitely smaller and smaller and we'll have infinitely

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more and more of them.

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So the width of this is dx.

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Let me draw it big, not so horrible looking.

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So if this is a sliver, it's width is dx.

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It's height is f of x,1.

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x,1 will be right in the center.

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And then it's distance from the center is of course x,1.

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Hopefully that make sense.

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So what's the volume of this shell?

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So the volume of the shell-- this shell, not this one-- the

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volume of the shell is going to be equal to the surface area

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of the shell times how wide that surface is.

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And that width is dx, so it's going to equal this times dx.

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So the volume of that shell is 2 pi x,1 times

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f of x,1 times dx.

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I think you see where I'm going with this now.

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So what would be the volume of the entire rotated

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figure, this thing here?

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Well I'm just going to sum up each of these shells.

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I have one shell there, then here I'll have a slightly less

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high shell, and up here I would have a much bigger shell,

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and I'll add them up.

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Here's one shell that goes around.

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Then they'll be another shell here, and I'll add them all up.

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And that's taking the integral.

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So the total volume of the figure when I rotate it around

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the y-axis is going to be-- and my boundary is from 0 to 1-- 2

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pi-- this one I just told you a particular x,1 but we're going

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to sum them over all of the x's.

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So it's going to be 2 pi x f of x dx.

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This is just a constant, so you could call it

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2 pi times x f of x.

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So let's take a particular example.

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Let's do it for x squared.

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Let's say the function is x squared.

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So in this case the volume is going to equal-- let's take the

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2 pi out-- 2 pi integral 0 to 1 x times f of x-- f of x in our

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case is x squared, which I drew earlier-- dx equals 2 pi.

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This is just x to the third, right?

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x to the third.

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So it's going to be 2 pi times the antiderivative

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of x to the third.

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Well that's x to the fourth over 4.

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Evaluate it at 1 minus evaluate it at 0.

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Well that equals 2 pi times 1 to the fourth is 1, so

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1/4 and then minus 0.

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So it's 2 pi times 1/4.

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So that's pi over 2.

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That's the volume, and we just rotated it around the y-axis.

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I will see you in the next video.

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