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Current time:0:00Total duration:9:24

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 double 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 the way I'm about to show you and maybe you won't see the difference or maybe you 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 I scrolled my drawing but anyway let's say we have a surface in three 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 this surface so this this whole area is is or this whole volume is what we're trying to figure out right between the dotted lines I think you can see it and 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 the XY plane and it's depending on how you view it this side of it is DX its length is DX and the tight 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 it's area the area of this little square is going to be DX times dy and then 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 there's going to be an approximation then we're going to take an infinite sum I think we're you know where this is going but let me do that let me 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 and intersects the top of the surface and the volume of this column not 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 bit small 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 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 all in in every different way so we could also have written this as f of X Y times DX times dy and of course since multiplication is associative I could have also written it as f of X Y times dy DX these are all equivalent and these all represent the volume of this column of that's 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 with a couple of things we could do we could add up all the volumes in the x-direction because you know between the the lower expound and the upper expound and then we'd have kind of a thin a thin sheet although it'll already have some depth because we're not adding up just DX is that 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 DX dy it would go back dy and come back here if we wanted to sum up all the the D X's and if we want to do that which impression would we use well we would 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 if we're summing with respect to X but we have the dy near first it's really not incorrect but it just becomes a little ambiguous are we something with respect to X or Y but here we could say okay 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 just write numbers here but I'm going to say well the lower bound here is X is equal to a X is equal to a and the upper bound here is X is equal to B X is equal to B and that will 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 on my video I think that's I think that's the newspaper people trying to sell me something anyway so once we have this 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 could integrate those we could add up the dys right because this with right here still dy we could add up all the different dy then we would have the volume of the whole figure so once we take this sum then we could take this sum where Y where Y is going from its bottom which we said with C from y is equal to C 2 y is 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 and though 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 let's start level da instead of going this way instead of summing up the D X's and getting this this sheet let's sum up the D Y's first right so we could take we're summing in the y direction first so we would get a sheet that's parallel to the y-axis now so it would the top of the sheet would look something like that right so if we're summing the dy is first we could take the sum we 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 integrating the order of integration now another way of saying this if 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 it the XY 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 + y times D a reason why I want to show you this is you see this in physics books all the time and I think it's 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 it's not obvious for me to know how to compute it actually is more confusing in it 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 be that it would be DX times dy or could either be dy times DX and when they do this double integral over domain that's the same thing as just adding up all of these squares where we do it here we're very ordered about it right we go it in the x-direction and then we add all of those up in the y-direction 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 domain it 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 could do it in pole coordinates you could do it a ton of different ways but I just wanted to show you that this is another way of having an intuition of the volume under surface and it's these are the exact same thing as this type of notation that you might see in a physics book sometimes it won't write a domain sometimes they'd write over a surface and will 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