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Let's now do another triple integral, and in this one I

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won't actually evaluate the triple integral.

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But what we'll do is we'll define the triple integral.

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We're going to something similar that we did in the

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second video where we figured out the mass

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using a density function.

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But what I want to do in this video is show you how to set

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the boundaries when the figure is a little

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bit more complicated.

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And if we have time we'll try to do it where we change

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the order of integration.

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So let's say I have the surface -- let me just make something

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up -- 2x plus 3z plus y is equal to 6.

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Let's draw that surface.

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It looks something like this.

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This will be my x-axis.

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This is going to be my z-axis.

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This is going to be my y-axis.

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Draw them out.

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x, y and z.

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And I care about the surface in the kind of positive octant,

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right, because when you're dealing with three-dimensionals

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we have, instead of four quadrants we have

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eight octants.

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But we want the octant where all x, y and z is positive,

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which is the one I drew here.

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So let's see, let me draw some -- what is the x-intercept?

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When y and z are 0, so we'll write here,

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that's the x-intercept.

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2x is equal to 6, so x is equal to 3.

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So 1, 2, 3.

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So that's the x-intercept.

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The y-intercept when x and z are 0 are on the y-axis,

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so y will be equal to 6.

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so we have 1, 2, 3, 4, 5, 6 is the y-intercept.

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Then finally the z-intercept when x and y are 0

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we're on the z-axis.

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3z will be equal to 6.

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So z is equal to 1, 2.

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So the figure that I care about will look something like

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this -- it's be this inclined surface.

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It will look something like that.

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In this positive octant.

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So this is the surface defined by this function.

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Let's say that I care about the volume, and I'm going

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to make it a little bit more complicated.

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We could say oh, well this was a volume between the

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surface and the xy plane.

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But I'm going to make it a little bit more complicated.

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Let's say the volume above this surface, and the

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surface z is equal to 2.

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So the volume we care about is going to look

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something like this.

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Let me see if I can pull off drawing this.

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If we go up 2 here -- let me draw the top in a different

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color, let me draw the top in green.

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So this is along the zy plane.

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And then the other edge is going to look

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something like this.

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Let me make sure I can draw it -- this is the hardest part.

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We'll go up 2 here, and this is along the zx plane.

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And we'd have another line connecting these two.

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So this green triangle, this is part of the

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plane z is equal to 2.

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The volume we care about is the volume between this top green

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plane and this tilted plane defined by 2x plus 3z

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plus y is equal to 6.

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So this area in between.

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Let me see if I can make it a little bit clearer.

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Because the visualization, as I say, is often the hardest part.

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So we'd have kind of a front wall here, and then the back

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wall would be this wall back here, and then there'd

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be another wall here.

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And then the base of it, the base I'll do in magenta

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will be this plane.

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So the base is that plane -- that's the bottom part.

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Anyway, I don't know if I should have made it that messy

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because we're going to have to draw dv's and d volumes on it.

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But anyway, let's try our best.

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So, if we're going to figure out the volume -- and actually,

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since we're doing a triple integral and we want to show

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that we have to do the triple integral, instead of doing a

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volume, let's do the mass of something of variable density.

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So let's say the density in this volume that we care about,

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the density function is a function of x, y and z.

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It can be anything.

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That's not the point of what I'm trying to teach here.

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But I'll just define something.

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Let's say it's x squared yz.

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Our focus is really just to set up the integrals.

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So the first thing I like to do is I visualize -- what we're

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going to do is we're going to set up a little cube in the

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volume under consideration.

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So if I had a -- let me do it in a bold color so that you can

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see it -- so if I have a cube -- maybe I'll do it in brown,

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it's not as bold but it's different enough from

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the other colors.

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So if I had a little cube here in the volume under

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consideration, that's a little cube -- you consider that dv.

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The volume of that cube is kind of a volume differential.

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And that is equal to dx -- no, sorry, this is dy.

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Let me do this in yellow, or green even better.

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So dy, which is this.

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dy times dx, dx times dz.

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That's the volume of that little cube.

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And if we wanted to know the mass of that cube, we would

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multiply the density function at that point times this dv.

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So the mass, you could call it d -- I don't know, dm.

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The mass differential is going to be equal to that times that.

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So x squared y z times this.

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dy, dx, and dz.

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And we normally switch this order around, depending on what

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we're going to integrate with respect to first so we

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don't get confused.

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So let's try to do this.

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Let's try to set up this integral.

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So let's do it traditionally.

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The last couple of triple integrals we did we integrated

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with respect to z first.

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So let's do that.

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So we're going to integrate with respect to z first. so

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we're going to take this cube and we're going to sum up all

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of the cubes in the z-axis.

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So going up and down first, right?

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So if we do that, what is the bottom boundary?

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So when you sum up up and down, these cubes are going to

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turn to columns, right?

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So what is the bottom of the column, the bottom bound?

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What's the surface?

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It's the surface defined right here.

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So, if we want that bottom bound defined in terms of

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z, we just have to solve this in terms of z.

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So let's subtract.

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So what do we get.

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If we want this defined in terms of z, we get 3z is

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equal to 6 minus 2x minus y.

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Or z is equal 2 minus 2/3x minus y over 3.

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This is the same thing as that.

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But when we're talking about z, explicitly defining a

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z, this is how we get it, algebraically manipulated.

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So the bottom boundary -- and you can visualize it, right?

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The bottom of these columns are going to go up and down.

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We're going to add up all the columns in up and

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down direction, right?

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You can imagine summing them.

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The bottom boundary is going to be this surface.

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z is equal to 2 minus 2/3x minus y over 3.

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And then what's the upper bound?

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Well, the top of the column is going to be this green

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plane, and what did we say the green plane was?

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It was z is equal to 2.

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That's this plane, this surface right here.

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Z is equal to 2.

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And, of course, what is the volume of that column?

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Well, it's going to be the density function, x squared yz

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times the volume differential, but we're integrating

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with respect to z first.

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Let me write dz there.

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I don't know, let's say we want to integrate with respect to --

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I don't know, we want to integrate with respect

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to x for next.

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In the last couple of videos, I integrated

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with respect to y next.

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So let's do x just to show you it really doesn't matter.

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So we're going to integrate with respect to x.

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So, now we have these columns, right?

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When we integrate with respect to z, we get the volume of each

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of these columns wher the top boundary is that plane.

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Let's see if I can draw it decently.

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The top boundary is that plane.

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The bottom boundary is this surface.

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Now we want to integrate with respect to x.

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So we're going to add up all of the dx's.

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So what is the bottom boundary for the x's?

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Well, this surface is defined all the way to -- the volume

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under question is defined all the way until x is equal to 0.

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And if you get confused, and it's not that difficult to get

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confused when you're imagining these three-dimensional things,

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say you know what, we already integrated with respect to z.

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The two variables I have left are x and y.

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Let me draw the projection of our volume onto the xy plane,

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and what does that look like?

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So I will do that.

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Because that actually does help simplify things.

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So if we twist it, if we take this y and flip it out like

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that, and x like that we'll get in kind of the traditional way

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that we learned when we first learned algebra.

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The xy-axis.

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So this is x, this is y.

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And this point is what?

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Or this point?

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What is that?

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That's x is equal to 3.

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So it's 1, 2, 3.

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That's x is equal to 3.

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And this point right here is y is equal to 6.

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So 1, 2, 3, 4, 5, 6.

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So on the xy-axis, kind of the domain -- you can view it that

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-- looks something like that.

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So one way to think about it is we've figured out if these

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columns -- we've integrated up/down or along the z-axis.

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But when you view it looking straight down onto it, you're

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looking on the xy plane, each of our columns are going to

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look like this where the column's going to pop out out

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of your screen in the z direction.

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But the base of each column is going to dx like that, and

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00:10:10,759 --> 00:10:13,039
then dy up and down, right?

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00:10:13,039 --> 00:10:15,799
So we decided to integrate with respect to x next.

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So we're going to add up each of those columns in the

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00:10:18,519 --> 00:10:20,860
x direction, in the horizontal direction.

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So the question was what is the bottom boundary?

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What is the lower bound in the x direction?

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00:10:24,879 --> 00:10:26,679
Well, it's x is equal to 0.

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If there was a line here, then it would be that line probably

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00:10:30,399 --> 00:10:34,740
as a function of y, or definitely as a function of y.

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So our bottom bound here is x is equal to 0.

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What's our top bound?

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I realize I'm already pushing.

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Well, our top bound is this relation, but it has to

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be in terms of x, right?

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So what is this relation.

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So, you could view it as kind of saying well, if z is equal

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to 0, what is this line?

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What is this line right here?

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So z is equal to 0.

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We have 2x plus y is equal to 6.

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We want the relationship in terms of x.

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So we get 2x is equal to 6 minus y where x is equal

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to 3 minus y over 2.

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And then finally we're going to integrate with respect to y.

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And this is the easy part.

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So we've integrated up and down to get a column.

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These are the bases of the column, so we've integrated

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in the x direction.

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Now we just have to go up and down with respect to y, or in

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the xy plane with respect to y.

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So what is the y bottom boundary?

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Well, it's 0.

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y is equal to 0.

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And the top boundary is y is equal to 6.

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And there you have it.

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We have set up the integral and now it's just a matter

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of chugging through it mechanically.

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But I've run out of time and I don't want this

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video to get rejected.

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00:11:44,659 --> 00:11:45,870
So I'll leave you there.

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00:11:45,870 --> 00:11:47,759
See you in the next video.