Double Integrals 4 Another way to conceptualize the double integral.
Double Integrals 4
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- I think it's very important to have as many ways as possible
- to view a certain type of problem, so I want to introduce
- you to a different way.
- Some people might have taught this first, but the way I
- taught it in the first integral video is kind of the way that I
- always think about when I do the problems.
- But sometimes, it's more useful to think about it the way I'm
- about to show you, and maybe you won't see the difference,
- or maybe you'll say, oh, Sal, those are just the
- exact same thing.
- Someone actually emailed me and told me that I should make it
- so I can scroll things, and I said, oh, that's not
- too hard to do.
- So I just did that, and I scrolled my drawing.
- But anyway, let's say we have a surface in 3 dimensions.
- It's a function of x and y.
- You give me a coordinate down here, and I'll tell you
- how high the surface is at that point.
- And we want to figure out the volume under that surface.
- We can very easily figure out the volume of a very small
- column underneath the surface.
- So this whole volume is what we're trying to figure out,
- right, between the dotted lines.
- I think you can see it.
- You have some experience visualizing this right now.
- So let's say that I have a little area here.
- We could call that da.
- Let me see if I can draw this.
- Let's say we have a little area down here, a little
- square in the x-y plane.
- And it's, depending on how you view it, this side of it is dx,
- its length is dx, and the height, you could say,
- on that side, is dy.
- Because it's a little small change in y there, and it's a
- little small change in x here.
- And its area, the area of this little square, is
- going to be dx times dy.
- And if we wanted to figure out the volume of the solid between
- this little area and the surface, we could just multiply
- this area times the function.
- Because the height at this point is going to be the
- value of the function, roughly, at this point.
- This is going to be an approximation, and then we're
- going to take an infinite sum.
- I think you know where this is going.
- But let me do that.
- Let me at least draw the little column that I want to show you.
- So that's one end of it, that's another end of it, that's the
- front end of it, that's the other end of it.
- So we have a little figure that looks something like that.
- A little column, right?
- It intersects the top of the surface.
- And the volume of this column, not too difficult.
- It's going to be this little area down here, which is,
- we could call that da.
- Sometimes written like that. da.
- It's a little area.
- And we're going to multiply that area times the height of
- this column, and that's the function at that point.
- So it's f of x and y.
- And of course, we could have also written it as, this
- da is just dx times dy, or dy times dx.
- I'm going to write it in every different way.
- So we could also have written this as f of
- xy times dx times dy.
- And of course, since multiplication is associative,
- I could have also written it as f of xy times dy dx.
- These are all equivalent, and these all represent the volume
- of this column, that's the between this little area
- here and the surface.
- So now, if we wanted to figure out the volume of the entire
- surface, we have a couple of things we could do.
- We could add up all of the volumes in the x-direction,
- between the lower x-bound and the upper x-bound, and then
- we'd have kind of a thin sheet, although it will already have
- some depth, because we're not adding up just the x's.
- There's also a dy back there.
- So we would have a volume of a figure that would extend from
- the lower x all the way to the upper x, go back dy,
- and come back here.
- If we wanted to sum up all the dx's.
- And if we wanted to do that, which expression would we use?
- Well, we would be summing with respect to x first, so we could
- use this expression, right?
- And actually, we could write it here, but it
- just becomes confusing.
- If we're summing with respect to x, but we have the
- dy written here first.
- It's really not incorrect, but it just becomes a little
- ambiguous, are we summing with respect to x or y.
- But here, we could say, OK.
- If we want to sum up all the dx's first, let's do that.
- We're taking the sum with respect to x, and let me, I'm
- going to write down the actual, normally I just write numbers
- here, but I'm going to say, well, the lower bound here is x
- is equal to a, and the upper bound here is x is equal to b.
- And that'll give us the volume of, you could imagine a
- sheet with depth, right?
- The sheet is going to be parallel to the x-axis, right?
- And then once we have that sheet, in my video, I think
- that's the newspaper people trying to sell me something.
- So once we have the sheet, I'll try to draw it here, too, I
- don't want to get this picture too muddied up, but once we
- have that sheet, then we can integrate those, we can
- add up the dy's, right?
- Because this width right here is still dy.
- We could add up of all the different dy's, and we
- would have the volume of the whole figure.
- So once we take this sum, then we could take this sum.
- Where y is going from it's bottom, which we said with c,
- from y is equal to c to y's upper bound, to y
- is equal to d.
- Fair enough.
- And then, once we evaluate this whole thing, we have the
- volume of this solid, or the volume under the surface.
- Now we could have gone the other way.
- I know this gets a little bit messy, but I think
- you get what I'm saying.
- Let's start with that little da we had originally.
- Instead of going this way, instead of summing up the dx's
- and getting this sheet, let's sum up the dy's first, right?
- So we could take, we're summing in the y-direction first.
- We would get a sheet that's parallel to the y-axis, now.
- So the top of the sheet would look something like that.
- So if we're coming the dy's first, we would take the sum,
- we would take the integral with respect to y, and it would be,
- the lower bound would be y is equal to c, and the upper
- bound is y is equal to d.
- And then we would have that sheet with a little depth, the
- depth is dx, and then we could take the sum of all of those,
- sorry, my throat is dry.
- I just had a bunch of almonds to get power to be able
- to record these videos.
- But once I have one of these sheets, and if I want to sum up
- all of the x's, then I could take the infinite sum of
- infinitely small columns, or in this view, sheets, infinitely
- small depths, and the lower bound is x is equal to a, and
- the upper bound is x is equal to b.
- And once again, I would have the volume of the figure.
- And all I showed you here is that there's two ways of doing
- the order of integration.
- Now, another way of saying this, if this little original
- square was da, and this is a shorthand that you'll see all
- the time, especially in physics textbooks, is that we
- are integrating along the domain, right?
- Because the x-y plane here is our domain.
- So we're going to do a double integral, a two-dimensional
- integral, we're saying that the domain here is two-dimensional,
- and we're going to take that over f of x and y times da.
- And the reason why I want to show you this, is you see this
- in physics books all the time.
- I don't think it's a great thing to do.
- Because it is a shorthand, and maybe it looks simpler, but for
- me, whenever I see something that I don't know how to
- compute or that's not obvious for me to know how to compute,
- it actually is more confusing.
- So I wanted to just show you that what you see in this
- physics book, when someone writes this, it's the exact
- same thing as this or this.
- The da could either be dx times dy, or it could either be dy
- times dx, and when they do this double integral over domain,
- that's the same thing is just adding up all of these squares.
- Where we do it here, we're very ordered about it, right?
- We go in the x-direction, and then we add all of those up in
- the y-direction, and we get the entire volume.
- Or we could go the other way around.
- When we say that we're just taking the double integral,
- first of all, that tells us we're doing it in two
- dimensions, over a domain, that leaves it a little bit
- ambiguous in terms of how we're going to sum
- up all of the da's.
- And they do it intentionally in physics books, because you
- don't have to do it using Cartesian coordinates,
- using x's and y's.
- You can do it in polar coordinates, you could do it
- a ton of different ways.
- But I just wanted to show you, this is another way to
- having an intuition of the volume under a surface.
- And these are the exact same things as this type of
- notation that you might see in a physics book.
- Sometimes they won't write a domain, sometimes they'd
- write over a surface.
- And we'll later do those integrals.
- Here the surface is easy, it's a flat plane, but sometimes
- it'll end up being a curve or something like that.
- But anyway, I'm almost out of time.
- I will see you in the next video.
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