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Welcome back.

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I don't know what I was thinking.

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Sometimes my brain malfunctions.

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But just going back to that problem we were doing, actually

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I think we should do it.

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I'm a little schizophrenic today.

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So let's figure out the equation for the

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volume of a sphere.

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So what's the equation?

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

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And let's just write y as a function of x, just so

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we can do it the way we did that last problem.

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So you get y squared is equal to r squared minus x squared.

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y is equal to the square root of r squared minus x squared.

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And let's draw it.

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So if this is my y-axis, this is my x-axis, and the

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equation-- draw it straight-- that's my x-axis, and then I

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actually have a circle tool, let me see if I can use it

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effectively-- well, close enough.

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There you go.

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I think you get the point.

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But anyway.

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This is just going to be the upper half of the circle.

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Actually, I should probably just undo that circle tool

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and try to draw it by hand.

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So y equals the square root of r squared minus x squared.

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That's just going to be the upper half of the circle.

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So it will be the positive x quadrant-- and then actually,

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I should have drawn the whole hemisphere.

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But anyway.

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Actually, let me do that, because I think it'll

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make-- edit, undo, let me clear all of this out.

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Sorry for wasting your time, but I think it'll be effective.

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OK.

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So let me redraw.

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So this, that's the y-axis, that's my x-axis, and then

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this-- the square root is, since it's a function, it can

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only have one value, so we assume it's defined as the

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positive square root.

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So if we were to graph that, it would look like this.

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Something like that, where this would be minus r and that's r.

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So if we want to find the volume of a sphere with radius

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r, we just have to rotate this function around the x-axis.

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This is the x-axis, that's the y-axis.

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So let's see what we can do.

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Let's just visualize the disks again.

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So let me make a disk.

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So let's say that that's the side of one of the disks again,

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and as we know, the depth of the disk is just

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going to be dx.

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That's how wide that disk is, dx.

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And its radius at any point is f of x, and in this case, it's

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y is equal to square root of r squared minus x squared.

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So what's the surface area of each disk?

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What's this?

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The surface area of each of the disks.

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I hope you know what I'm saying.

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So area is equal to pi r squared, the radius at any

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point is equal to this, radius is equal to y which is equal to

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square root-- and remember, this is not this r.

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This is the radius of this disk.

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I know it might be a little confusing.

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y is equal to the square root of r squared minus x squared.

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So the area is going to equal pi times this squared.

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So if you square this quantity, you just get rid of the

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square root sign, right?

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So pi r squared minus x squared, and that's the

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area, and so what's the volume of that disk?

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Well just like we've done in every video up to this point,

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the volume of that disk is just that, so the volume of that

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disk is just this pi r squared minus x squared times dx.

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And so if we want to figure out the volume of all these disks,

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I have a disk here, a disk here, going around and around

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and around and around and around and they get smaller and

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smaller until we have a sphere.

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We just take the integral, the upper bound is positive r, the

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lower bound is minus r, and we take the integral of

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this expression.

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pi-- let me distribute it, because that's going to make it

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easier-- pi r squared, which is just a constant term, minus pi

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x squared, all of that dx.

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So what's the antiderivative of that expression?

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The antiderivative within the parentheses.

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Well, this is just a constant term.

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pi r squared, that's just a number, because we're just

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taking the integral with respect to x.

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So the antiderivative of pi r squared is just pi r squared x,

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the derivative of pi r squared x is just pi r squared, minus--

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and we did this in the last video.

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Actually, well now, it's the antiderivative x squared, which

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is x to the third over 3, and the pi is just a constant, so

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pi x to the third over 3, and we're going to evaluate

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that at r and minus r.

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Let me erase some stuff, looks like I'm running out of space.

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Hopefully all of that you know by now.

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OK, back to the pen tool.

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OK.

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So let's evaluate it at r.

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So this is pi r squared, and then for x, we'll substitute

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the positive r times r minus pi x cubed, but now we have this r

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here, so r cubed over 3 minus pi r squared, and then we have

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a minus r here, because we're evaluating the antiderivative

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at minus r, times minus r, minus pi minus r cubed.

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So what's minus r cubed?

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It's r cubed, but we'll keep the minus sign.

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r cubed, and at that minus sign, let's just make

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that-- that'll turn that into a plus-- over 3.

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Let's see if we can clean this up a little bit.

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So that first term is pi r cubed, r squared times r, minus

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essentially 1/3pi r cubed.

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And then, what is this?

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This is pi r cubed, but then we have a minus sign up.

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This is minus pi r cubed, and then we have a minus sign up

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here, so this becomes plus pi r cubed, and then minus-- because

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we have a plus here and a minus out here, so distribute it--

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so minus 1/3pi r cubed.

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And let's see, what do we have?

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We have essentially 1.

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If we just distribute out the pi r cubes, we have pi r cubed

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times 1 minus 1/3 plus 1 minus 1/3.

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Well that's 2 minus 2/3, or another way, let's see, is 2

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minus 2/3-- this is turning into a fractions problem-- and

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what's-- well, that's 6/3 minus 2/3, it equals 4/3.

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So this is equal to 4/3.

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So the volume of the sphere is equal to 4/3 pi r cubed, which

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is the equation for the volume of a sphere.

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And actually, now that I realize, it did take

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me eight and a half minutes, so I am glad.

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My first intuition is always correct, I am glad I did

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this in a separate video.

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But that should be pretty interesting to you.

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And it makes a lot of sense.

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It's going to be a cube of the radius, pi is involved.

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The 4/3 is interesting, just in terms of how it relates

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to everything else.

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Area is pi r squared, and then all of a sudden you get a 4/3

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here, so it is something for you to think about.

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Anyway, hopefully you found that fun.

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I'll see you in the next video.

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