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Double integrals 4

Another way to conceptualize the double integral. Created by Sal Khan.

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  • mr pants teal style avatar for user Michael Jerzy Regdosz
    Where do I get these almonds for power?
    (50 votes)
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  • leaf green style avatar for user Vu
    Instead of going from infinitely thin lines to infinitely thin papers and finally to the whole volume, we're going from relatively larger sticks, line up those stick to get a relatively thicker mattress, then pile up those mattresses and get the volume.

    It's the same thing, except in the other way we deal with "smaller" items (but this doesn't matter since the values of those items become infinitely small so they can't be compared).

    Am I missing something?
    (24 votes)
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  • blobby green style avatar for user rhys_hirang
    Does Sal have any videos that integrate using polar coordinates?
    (13 votes)
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  • old spice man green style avatar for user Les
    At , Sal mentions that dA is used as a shorthand. Isn't it also used to generalize dxdy so that the order of integration (and the coordinate system) is not specified? I feel like this would make it more applicable in proofs and theorems and such.
    (8 votes)
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  • male robot donald style avatar for user Nikhil Prabhu
    In-spite of the approach to the problem being different from the first video. The way Sal goes about solving it is exactly the same. Is it only a way to visualise the problem differently? (since the volume of a column is different from a sheet etc.)
    (3 votes)
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    • starky ultimate style avatar for user blaise.brignac
      Pretty much. In the first video it was cross sections of sheets intersecting, but in this video it's going strait to the matter of taking infinitely many columns of the space under the graph. They both get you the end, but take a slightly different point of view.
      (5 votes)
  • leaf green style avatar for user Rebekah Love
    Question: Do I always need to draw the two dimensional graph of the bounds? Or can I jump into integrating? Do I NEED to do this for the limits of integration? I'm not opposed to it, I Just prefer a mathematical way so that I don't have to think about how to draw it or take the time to do it on my calculator. Do I just solve for things in terms of x when I do dydx, and vice versa? And my teacher said you can't always just choose which order you want to do it. How do I know when I can and when I can't? Oh, and can you do one on polar coordinates? I'm having trouble finding the limits of integration for theta =/
    (3 votes)
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  • hopper jumping style avatar for user Hackmous
    i can`t understand what is the difference between simple integration and double integration,,in simple one we used to find area under the curve and in this(double integration) volume under the curve,,, so please explain me the difference b/w them,,maybe i am missing something
    (3 votes)
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    • leaf blue style avatar for user Stefen
      I don't think you are missing anything at this stage. The only comment I have is that, yes, single integration is the area under a curve, whereas the double is volume under a surface (not curve as you wrote) if you are integrating a function such that z=f(x, y) (Remember - there is no volume under a curve - why?).

      Is there something in particular you think should be different?

      When you get to triple integrals, there is also a volume relationship, but there is also much more depending on the function being integrated - so the interpretation of the result of triple integrals can be a bit more complicated.
      (3 votes)
  • leaf green style avatar for user Deep Mistry
    what is double integration of cos(x+y)dx.dy first integrating from pi/2 to x and second integration from 0 to pi/2 ?
    (2 votes)
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  • blobby green style avatar for user 077bce142.sansar
    I didn't knew in America, Europe education is this clear concept. I wish I would have done bachelor's degree in USA.

    CAn you make video on double integral in polar just like the first method ? So we could visualize ??
    (2 votes)
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  • aqualine tree style avatar for user Steve Wallace
    what is the key point I seem to be missing with calculating the volume under a surface defined as Z=f(x,y), and above an area in the xy plane using double integrals or triple integrals. I've seen examples where I can't see the difference between the two, yet one uses a double and the other a triple?
    (2 votes)
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    • leaf blue style avatar for user Stefen
      (any chance you could send a reference to the video/problems you mention?)

      It is easy to set up a double integral of the form z=f(xy) into a triple integral where the bounds of z are 0 from below and the function f(x,y) above: ∫∫f(x,y)dydx = ∫∫∫dzdydx <from z=0 to z=f(xy)>
      Notice that the volume is between an arbitrary surface z=f(x,y) and the plane z=0.

      With the triple integral we can find the volume between two arbitrary surfaces, and moments and centroids etc. of the volume if given the appropriate function, eg ∫∫∫f(x,y,z).
      (1 vote)

Video transcript

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. So. 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. Right? 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. Right? Because the height at this point is going to be the value of the function, roughly, at this point. Right? 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. Anyway. 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.