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I think it's very important to have as many ways as possible

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to view a certain type of problem, so I want to introduce

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you to a different way.

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Some people might have taught this first, but the way I

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taught it in the first integral video is kind of the way that I

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always think about when I do the problems.

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But sometimes, it's more useful to think about it the way I'm

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about to show you, and maybe you won't see the difference,

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or maybe you'll say, oh, Sal, those are just the

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exact same thing.

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Someone actually emailed me and told me that I should make it

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so I can scroll things, and I said, oh, that's not

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too hard to do.

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So I just did that, and I scrolled my drawing.

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But anyway, let's say we have a surface in 3 dimensions.

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It's a function of x and y.

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You give me a coordinate down here, and I'll tell you

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how high the surface is at that point.

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And we want to figure out the volume under that surface.

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So.

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We can very easily figure out the volume of a very small

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column underneath the surface.

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So this whole volume is what we're trying to figure out,

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right, between the dotted lines.

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I think you can see it.

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You have some experience visualizing this right now.

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So let's say that I have a little area here.

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We could call that da.

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Let me see if I can draw this.

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Let's say we have a little area down here, a little

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square in the x-y plane.

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And it's, depending on how you view it, this side of it is dx,

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its length is dx, and the height, you could say,

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on that side, is dy.

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Right?

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Because it's a little small change in y there, and it's a

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little small change in x here.

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And its area, the area of this little square, is

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going to be dx times dy.

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And if we wanted to figure out the volume of the solid between

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this little area and the surface, we could just multiply

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this area times the function.

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Right?

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Because the height at this point is going to be the

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value of the function, roughly, at this point.

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Right?

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This is going to be an approximation, and then we're

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going to take an infinite sum.

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I think you know where this is going.

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But let me do that.

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Let me at least draw the little column that I want to show you.

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So that's one end of it, that's another end of it, that's the

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front end of it, that's the other end of it.

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So we have a little figure that looks something like that.

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A little column, right?

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It intersects the top of the surface.

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And the volume of this column, not too difficult.

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It's going to be this little area down here, which is,

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we could call that da.

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Sometimes written like that. da.

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It's a little area.

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And we're going to multiply that area times the height of

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this column, and that's the function at that point.

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So it's f of x and y.

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And of course, we could have also written it as, this

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da is just dx times dy, or dy times dx.

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I'm going to write it in every different way.

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So we could also have written this as f of

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xy times dx times dy.

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And of course, since multiplication is associative,

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I could have also written it as f of xy times dy dx.

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These are all equivalent, and these all represent the volume

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of this column, that's the between this little area

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here and the surface.

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So now, if we wanted to figure out the volume of the entire

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surface, we have a couple of things we could do.

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We could add up all of the volumes in the x-direction,

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between the lower x-bound and the upper x-bound, and then

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we'd have kind of a thin sheet, although it will already have

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some depth, because we're not adding up just the x's.

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There's also a dy back there.

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So we would have a volume of a figure that would extend from

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the lower x all the way to the upper x, go back dy,

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and come back here.

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If we wanted to sum up all the dx's.

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And if we wanted to do that, which expression would we use?

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Well, we would be summing with respect to x first, so we could

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use this expression, right?

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And actually, we could write it here, but it

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just becomes confusing.

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If we're summing with respect to x, but we have the

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dy written here first.

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It's really not incorrect, but it just becomes a little

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ambiguous, are we summing with respect to x or y.

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But here, we could say, OK.

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If we want to sum up all the dx's first, let's do that.

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We're taking the sum with respect to x, and let me, I'm

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going to write down the actual, normally I just write numbers

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here, but I'm going to say, well, the lower bound here is x

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is equal to a, and the upper bound here is x is equal to b.

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And that'll give us the volume of, you could imagine a

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sheet with depth, right?

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The sheet is going to be parallel to the x-axis, right?

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And then once we have that sheet, in my video, I think

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that's the newspaper people trying to sell me something.

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Anyway.

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So once we have the sheet, I'll try to draw it here, too, I

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don't want to get this picture too muddied up, but once we

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have that sheet, then we can integrate those, we can

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add up the dy's, right?

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Because this width right here is still dy.

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We could add up of all the different dy's, and we

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would have the volume of the whole figure.

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So once we take this sum, then we could take this sum.

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Where y is going from it's bottom, which we said with c,

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from y is equal to c to y's upper bound, to y

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is equal to d.

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Fair enough.

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And then, once we evaluate this whole thing, we have the

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volume of this solid, or the volume under the surface.

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Now we could have gone the other way.

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I know this gets a little bit messy, but I think

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you get what I'm saying.

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Let's start with that little da we had originally.

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Instead of going this way, instead of summing up the dx's

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and getting this sheet, let's sum up the dy's first, right?

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So we could take, we're summing in the y-direction first.

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We would get a sheet that's parallel to the y-axis, now.

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So the top of the sheet would look something like that.

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So if we're coming the dy's first, we would take the sum,

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we would take the integral with respect to y, and it would be,

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the lower bound would be y is equal to c, and the upper

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bound is y is equal to d.

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And then we would have that sheet with a little depth, the

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depth is dx, and then we could take the sum of all of those,

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sorry, my throat is dry.

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I just had a bunch of almonds to get power to be able

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to record these videos.

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But once I have one of these sheets, and if I want to sum up

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all of the x's, then I could take the infinite sum of

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infinitely small columns, or in this view, sheets, infinitely

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small depths, and the lower bound is x is equal to a, and

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the upper bound is x is equal to b.

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And once again, I would have the volume of the figure.

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And all I showed you here is that there's two ways of doing

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the order of integration.

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Now, another way of saying this, if this little original

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square was da, and this is a shorthand that you'll see all

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the time, especially in physics textbooks, is that we

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are integrating along the domain, right?

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Because the x-y plane here is our domain.

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So we're going to do a double integral, a two-dimensional

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integral, we're saying that the domain here is two-dimensional,

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and we're going to take that over f of x and y times da.

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And the reason why I want to show you this, is you see this

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in physics books all the time.

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I don't think it's a great thing to do.

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Because it is a shorthand, and maybe it looks simpler, but for

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me, whenever I see something that I don't know how to

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compute or that's not obvious for me to know how to compute,

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it actually is more confusing.

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So I wanted to just show you that what you see in this

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physics book, when someone writes this, it's the exact

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same thing as this or this.

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The da could either be dx times dy, or it could either be dy

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times dx, and when they do this double integral over domain,

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that's the same thing is just adding up all of these squares.

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Where we do it here, we're very ordered about it, right?

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We go in the x-direction, and then we add all of those up in

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the y-direction, and we get the entire volume.

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Or we could go the other way around.

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When we say that we're just taking the double integral,

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first of all, that tells us we're doing it in two

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dimensions, over a domain, that leaves it a little bit

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ambiguous in terms of how we're going to sum

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up all of the da's.

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And they do it intentionally in physics books, because you

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don't have to do it using Cartesian coordinates,

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using x's and y's.

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You can do it in polar coordinates, you could do it

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a ton of different ways.

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But I just wanted to show you, this is another way to

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having an intuition of the volume under a surface.

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And these are the exact same things as this type of

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notation that you might see in a physics book.

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Sometimes they won't write a domain, sometimes they'd

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write over a surface.

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And we'll later do those integrals.

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Here the surface is easy, it's a flat plane, but sometimes

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it'll end up being a curve or something like that.

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But anyway, I'm almost out of time.

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I will see you in the next video.

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