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So far, we've used integrals to figure out the

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area under a curve.

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And let's just review a little bit of the intuition, although

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this should hopefully be second nature to you at this point.

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If it's not, you might want to review the definite

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integration videos.

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But if I have some function-- this is the xy plane, that's

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the x-axis, that's the y-axis-- and I have some function.

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Let's call that, you know, this is y is equal to

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some function of x.

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Give me an x and I'll give you a y.

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If I wanted to figure out the area under this curve, between,

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let's say, x is equal to a and x is equal to b.

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So this is the area I want to figure out.

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What I do is, I split it up into a bunch of columns

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or a bunch of rectangles.

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Where-- let me draw one of those rectangles-- where you

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could view-- and there's different ways to do this,

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but this is just a review.

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Where you could review-- that's maybe 1 of the rectangles.

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Well, the area of the rectangle is just base

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times height, right?

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Well, we're going to make these rectangles really skinny and

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just sum up an infinite number of them.

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So we want to make them infinitely small.

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But let's just call the base of this rectangle dx.

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And then the height of this rectangle is going to be

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f of x, at that point.

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It's going to be f of-- if this is x0, or whatever, you can

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just call it f of x, right?

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That's the height of that rectangle.

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And if we wanted to take the sum of all of these

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rectangles-- right?

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There's just going to be a bunch of them.

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One there, one there.

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Then we'll get the area, and if we have infinite number of

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these rectangles, and they're infinitely skinny, we

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have exactly the area under that curve.

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That's the intuition behind the definite integral.

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And the way we write that-- it's the definite integral.

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We're going to take the sums of these rectangles, from x is

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equal to a, to x is equal to b.

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And the sum, or the areas that we're summing up, are going to

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be-- the height is f of x, and the width is d of x.

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It's going to be f of x times d of x.

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This is equal to the area under the curve.

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f of x, y is equal to f of x, from x is equal to

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a to x is equal to b.

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And that's just a little bit of review.

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But hopefully, you'll now see the parallel of how we

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extend this to taking the volume under a surface.

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So first of all, what is a surface?

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Well, if we're thinking in three dimensions, a

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surface is going to be a function of x and y.

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So we can write a surface as, instead of y is a function

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of f and x-- I'm sorry.

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Instead of saying that y is a function of x, we can write a

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surface as z is equal to a function of x and y.

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So you can kind of view it as the domain.

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Right?

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The domain is all of the set of valid things that you

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can input into a function.

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So now, before, our domain was just-- at least, you know, for

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most of what we dealt with-- was just the x-axis, or kind of

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the real number line in the x direction.

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Now our domain is the xy plane.

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We can give any x and any y-- and we'll just deal with the

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reals right now, I don't want to get too technical.

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And then it'll pop out another number, and if we wanted to

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graph it, it'll be our height.

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And so that could be the height of a surface.

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So let me just show you what a surface looks like, in

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case you don't remember.

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And we'll actually figure out the volume under this surface.

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So this is a surface.

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I'll tell you its function in a second, but it's

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pretty neat to look at.

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But as you can see, it's a server.

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It's like a piece of paper that's bent.

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Let's see, let me rotate it to its traditional form.

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So this is the x direction, this is the y direction.

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And the height is a function of where we are in the xy plane.

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So how do we figure out the volume under a

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surface like this?

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How do we figure out the volume?

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It seems like a bit of a stretch, given what we've

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learned from this.

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But what if-- and I'm just going to draw an abstract

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surface here-- let me draw some axis.

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Let's see, that's my x-axis.

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That's my y-axis.

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That's my z-axis.

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I don't practice these videos ahead of time, so I'm often

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wondering what I'm about to draw.

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OK.

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So that's x, that's y, and that's z.

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And let's say I have some surface.

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I'll just draw something.

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I don't know what it is.

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Some surface.

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This is our surface.

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z is a function of x and y.

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So, for example, you give me a coordinate in the xy plane.

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Say, here, I'll put it into the function and it'll

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give us a z value now.

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And I'll plot it there and it'll be a point

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on the surface.

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So what we want to figure out is the volume

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under the surface.

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And we have to specify bounds, right?

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From here, we said x is equal to a, to x is equal to b.

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So let's make a square bound first, because this

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keeps it a lot simpler.

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So let's say that the domain or the region-- not the domain--

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the region of-- the x and y region of this part of the

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surface under which we want to calculate the volume.

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Let's say, the shadow-- if the sun was right above

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the surface, the shadow would be right there.

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Let me try my best to draw this neatly.

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So this is what we're going to try to figure

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out the volume of.

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And let's say-- so, if we wanted to draw it in the xy

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plane, like you can kind of view the projection of the

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surface of the xy plane, or the shadow of the

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surface of the xy plane.

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What are the bounds?

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You can almost view-- what are the bounds of the domain?

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Well, let's say that this point-- let's say that this

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right here, that's 0, 0 in the xy plane.

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Let's say that this is y is equal to-- I don't know,

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that's y is equal to a.

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That's this line right here.

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Y is equal to a.

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And let's say that this line right here is x is equal to b.

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Hope you get that, right?

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This is the xy plane.

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If we have a constant x, it would be a line like that.

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A constant y, a line like that.

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And then we have the area in between it.

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So how do we figure out the volume under this?

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Well, if I just wanted to figure out the area of--

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let's just say, this sliver.

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Let's say we had a-- well, actually let

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me go the other way.

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Let's say we had a constant y.

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So let's say I had some sliver.

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I don't want to confuse you.

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Let's say that I had some constant y.

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I just want to give you the intuition.

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You know, let's say.

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I don't know what that is.

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It's an arbitrary y.

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But for some constant y, what if I could just figure out the

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area under the curve there?

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How would I figure out just the area under that curve?

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It'll be a function of which y I pick, right?

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Because if I pick a y here, it'll be a different area.

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If I pick a y there, it'll be a different area.

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But I could view this now as a very similar problem

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to this one up here.

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I could have my dx's-- let me pick a vibrant color

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so you can see it.

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Let's say that's dx, right?

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That's a change in x.

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And then the height is going to be a function of the x

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I have and the y I picked.

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Although I'm assuming, to some degree, that

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that's a constant y.

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So what would be the area of this sheet of paper?

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It's kind of a constant y.

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It's part of-- it's a sheet of paper within this volume,

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you can kind of view it.

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Well, it would be-- we said the height of each of these

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rectangles is f of xy, right?

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That's the height.

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It depends which x and y we pick down here.

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And then its width is going to be d of x.

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Not d of x, dx.

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And then if we integrated it, from x is equal to 0, which was

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back here, all the way to x is equal to b, what

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would it look like?

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It would look like that. x is going from 0 to b.

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Fair enough.

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And this would actually give us a function of y.

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This would give us an expression so that I would know

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the area of this kind of sliver of the volume, for any

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given value of y.

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If you give me a y, I can tell you the area of the sliver

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that corresponds to that y.

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Now what could I do?

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If I know the area of any given sliver, what if I multiply the

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area of that sliver times dy?

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This is a dy.

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Let me do it in a vibrant color.

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So dy, a very small change in y.

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If I multiplied this area times a small dy, then

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all of a sudden I have a sliver of volume.

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Hopefully that makes some sense.

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I'm making that-- that little cut that I took the area of--

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by making it three dimensional.

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So what would be the volume of that sliver?

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The volume of that sliver will be this function of y times dy,

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or this whole thing times dy.

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So it would be the integral from 0 to b of f of xy dx.

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That gives us the area of this blue sheet.

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Now if I multiply this whole thing times dy,

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I get this volume.

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It gets some depth.

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This little area that I'm shading right here gets

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depth of that sheet.

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Now if I added all of those sheets that now have depth, if

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I took the infinite sum-- so if I took the integral of this

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from my lower y bound-- from 0 to my upper y bound, a, then--

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at least based on our intuition here-- maybe I will have

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figured out the volume under this surface.

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But anyway, I didn't want to confuse you.

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But that's the intuition of what we're going to do.

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And I think you're going to find out that actually

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calculating the volumes are pretty straightforward,

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especially when you have fixed x and y bounds.

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And that's what we're going to do in the next video.

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See you soon.

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