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Welcome back.

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On the last video we came to the conclusion that we could

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figure out the volume when we rotate a function about the

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x-axis, so let's apply that to an actual exercise.

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I'm going to erase everything because I don't want you to

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memorize this, because frankly I haven't memorized this.

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And if you do, you'll forget it one day and then you

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won't know how to do it.

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But if you understand why it works then you'll never forget.

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As long as you remember basic integration.

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Maybe you want to memorize it if your teacher tends to give

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you a test that don't have much extra time in them, just

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to speed up the process.

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But you should know what's going on.

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So let me draw the axes again.

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That's my y-axis.

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That's my x-axis.

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And so since our first example was y equals square root of

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x, let's stick with that.

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And for reasons that might become apparent that tends to

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be one of the more typical examples when you rotate

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things around axes.

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So let me see if I can draw it as well as I drew it last time.

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Almost.

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OK, so that's y equals square root of x, it's just f of x

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this time, I've defined it.

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This is the x-axis.

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That's the y-axis.

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And I'm going to rotate this around the x-axis again.

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So I'm going to get a sideways looking cup thing.

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And let's say I want to figure out the volume of that cup

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between the points 0-- and to make it simple, let's just

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say the points 0 and 1.

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So essentially we're just going to get a cup, a sideways

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cup, it's going to look something like this.

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It's going to look something like this.

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That's going to be the-- that's a horrible-- the

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opening of the cup.

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Actually why don't I use the circle tool.

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It just dawned on me that I had a circle tool.

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So the opening of the cup will look like that.

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Actually I could draw it right here.

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This would be the opening of the cup.

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Well-- you can see sometimes that my videos are

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a little unplanned.

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There you go.

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So that would be the opening of the cup.

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This is excellent.

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This tool is very well suited for what I'm doing here.

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We're rotating around that way.

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We're turning that function.

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So the cup's going to look like that, so the bottom part of the

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cup's going to look like this.

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And it's solid, so we want the volume of the whole thing.

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In future videos I'm going to show you actually how to figure

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out the surface area of the cup, which I find in some

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ways more interesting.

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So how do we think about that again?

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Let's just rederive it, but this time we'll use

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a specific equation.

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So we just have to figure out what is the volume of one disk

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and then sum up all the disks.

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So let's say this disk right here-- actually let's just take

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this disk at the end point right here that I've already

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drawn something for.

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So what's the radius of this disk?

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The radius of that disk is f of x at that point.

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Well f of x at that point is just square root of x.

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Radius is equal to square root of x.

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And so the area of that disk is going to equal pi r squared.

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Well the radius is square root of x, so it equals pi times

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square root of x squared.

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So it equals pi times x.

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That's the area of each disk.

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And then if we want the volume, you just have to multiply the

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area of that surface times the depth of the disk.

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I'm just trying to show.

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You can imagine that this is kind of like a quarter and this

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is the side of the quarter.

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We saw in the last video that depth, that's just a very

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small change in x, because we want each disk to be

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infinitesimally thin.

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So the width is just dx at any point.

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So the volume of each disk is equal to the area, which we

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just figured out was pi x times the depth, times dx.

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That's the volume of each disk.

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So the total volume is going to be equal to the

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sum of all of these.

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That was one disk I drew, then you're going to have another

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one here, you're going to have another one here,

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another one here.

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You're going to have infinitely many, and you want them to be

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super, super, super thin so that you get an accurate

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measure of the exact volume of this curve.

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Otherwise it would just be an approximation, and that's

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where we use the integral.

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So it will be the integral from.

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And my original boundaries were 0 to 1.

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The disk we used as an example, this is probably you know the

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last disk, so this one will actually have a radius of the

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square root of 1, which is 1.

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Not that you have to know that, I'm just trying to keep

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emphasizing the visualization.

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So what will be the integral?

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Well we're going to go from 0 to 1, and we're going to sum up

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a bunch of these disks, which we've already defined,

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so it's pi x dx.

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This is looking to be a fairly straightforward integral.

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So what's the integral of that?

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Pi is just a constant and the antiderivative of x is x to

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the 1/2 over-- I'm sorry.

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It's x squared over 2.

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I've been a little rusty since I last did some

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antiderivatives.

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So we get x squared, we get pi times x squared over 2.

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That's the antiderivative of that.

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And then we have to evaluate it at 1 and then subtract

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it and evaluate it at 0.

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And so what do we have.

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We get 1/2 pi, so we get pi over 2 minus 0 pi, minus 0.

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So it equals pi over 2.

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There we go.

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We just figured out the volume of this cup from 0 to 1.

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Reasonably interesting.

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Let's see if we can do that again to figure out the-- just

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to give you another example, just hit the point home-- to

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see if we can figure out the volume of a sphere,

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the equation for the volume of a sphere.

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So what's the equation for a circle?

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It's x squared plus y squared is equal to r squared.

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And let's write that in terms of y is a function of x, just

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so we have something that we can work with the

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way we learned it.

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So we get y squared is equal to r squared minus x squared, and

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then we get y is equal to the square root of r squared

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minus x squared.

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Actually now that I realize it, I'm going to not do this,

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because I think I'm going into too complicated a problem.

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I did that on a fly.

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But in the next video I will do slightly more complicated

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without going to this one, because I probably don't have

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time for it I just realized.

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Anyway I'll see you in the next video.

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