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# Double integral 1

## Video transcript

so far we've used integrals to figure out the area under curve and let's just review a little bit of the intuition although this should hopefully be second nature to you at this point if it's not you might want to review the definite integration videos but if I have some function this is the xy-plane that's the x-axis that's the y-axis and I have some function let's call that you know this is y is equal to some function of X you give me an X and I'll give you a Y if I wanted to figure out the area under this curve between let's say X is equal to a and X is equal to B so this is the area I want to figure out this area right here what I do is I split it up into a bunch of into a bunch of columns or a bunch of rectangles where let me draw one of those rectangles where you could view and there's different ways to do this but this is just a review where you could review that's maybe one of the rectangles well the area of the rectangle is just base times height right we're going to make these rectangles really skinny just sum up an infinite number of them so we want to make them infinitely small but let's just call the base of this rectangle DX and then the height of this rectangle is going to be f of X at that point right it's going to be F of if this is X naught or whatever you can just call it f of X right that's the height of that rectangle and if we wanted to take the sum of all of these rectangles right there's just going to be a bunch of them one there one there then we'll get the area and if these if we have infinite number of these rectangles that they're infinitely skinny we have exactly the area under that curve that's the intuition behind the definite integral and the way we write that it's the definite integral we're going to take the sums of these rectangles from X is equal to a to X is equal to B and the sum or the areas that we're summing up are going to be the height is f of X and the width is d of X it's going to be f of x times d of X this is equal to the area under the curve f of XY is equal to f of X from X is equal to 8 X is equal to B and that's just a little bit of review but hopefully you'll now see the parallel of how we extend this to taking the volume under a surface so first of all what is the surface well if we're thinking in three dimensions the surface is going to be a function of f of X and Y so we can write a surface as instead of Y is a function of F and X I'm sorry instead of saying that Y is a function of X we can write a surface as Z is equal to a function of x and y so you can kind of view it as the domain right the domain is all of the set of valid things that you can input into a function so now before our domain was just at least you know for most of what we dealt with was just the x-axis or kind of the real number line in the X direction now our domain is the XY plane we can given any X and any Y and well we'll just deal with reals right now but I want to get to technical and then it'll pop out another number and if we wanted to graph it will be our height and so that could be the height of a surface so let me just show you what a surface looks like in case you don't remember and we'll actually figure out the volume under this surface so this is a surface I'll tell you its function in a second but it's pretty neat to look at but as you can see it's a surface it's like it's like a piece of paper that's bent right let's see let me rotate it to this traditional form so this is this is the X Direction this is the Y direction and the height is a function of where we are in the XY plane so how do we figure out the volume under a surface like this right how do we figure out the volume it seems like a bit of a stretch given what we've learned from this but what if and I'm just going to draw an abstract surface here let me draw some axes let's say that's my x-axis that's my y-axis to my z-axis I don't practice these videos ahead of time so I'm often wondering what I'm about to draw okay so that's X that's why and that's Z and let's say I have some surface some surface I'll just draw something I don't know what it is some surface this is this is our surface Z is a function of x and y so you give me a coordinate in the XY plane say here I'll put it into the function and it'll give us a Z value and I would plot it there and it'll be a point on the surface so what we want to figure out is a volume under the surface and we have to specify bounds right from here we said X is equal to a to X is equal to B so let's make a square bound first because this keeps it a lot simpler so let's say that the the domain or the region not the domain the region of the X&Y region of this part of the of the surface under which we want to calculate the volume let's say it's let's say the shadow we if the Sun was right above the surface the shadow would be right there let me try my best to draw this neatly so this is what we're going to try to figure out the volume of all right and let's say let's say so if we wanted to draw it in the XY plane like just you can kind of view the projection of the surface in the XY plane or the shadow of the surface in the XY plane what are the bounds oh you can almost view it what are the bounds of the domain well let's say that this point let's say that this this right here that's 0 0 and the XY plane let's say that this is Y is equal to I don't know that's Y is equal to ay that's this line right here Y is equal to a and let's say that this line right here is X is equal to B hope you get that right this is the XY plane if you have a constant X it would be aligned like that a constant Y aligned like that and then we have the area in between it so how do we figure out the volume under this well if I just wanted to figure out the area of a of let's just say this sliver let's say we we had a well actually let go the other way let's say we had a constant Y so let's say for any I had some sliver I don't want to confuse you let's say that I have some constant Y now I just want to give you the intuition you know let's say I don't know what that is just an arbitrary Y but for some constant Y what if I could just figure out the area under the curve there right how would I figure out just the area under that curve it'll be a function of which why I pick right because if I pick a y here it'll be a different area if I pick a Y there it'll be a different area but I could view this now as a very similar problem to this one up here I could have my DX is let me pick a vibrant color so you can see it let's say that's DX right that's a change in X and then the height this height is going to be that height it's going to be a function of the X I have and the end end and and the y I've picked right although I'm assuming to some degree that that's a constant Y so what would be what would be the area of this sheet of paper right it's kind of a constant Y it's it's part of it's a sheet of paper within this volume you can kind of view it well it would be we set the height of each of these rectangles is f of XY f of XY right that's the height depends which x and y we pick down here and then its width is going to be d of X not D of X DX and then if we integrated it from X is equal to zero which was back here all the way to X is equal to B what would it look like it look like that X is going from zero to be fair enough and this would actually give us a function of Y this would give us an expression so that if it I would know the area of this of this kind of sliver of the volume for any given value of y if you give me a Y I can tell you the area of the sliver that corresponds to that Y now what can I do if I know the area of any given sliver what if I multiply the area of that sliver times dy right this is a dy let me do it in a in a vibrant color so dy a very small change in Y right if I multiply this area times a small dy then all of a sudden I have a sliver of volume hopefully that makes some sense right I'm making that that little cut that I that I took the area of I'm making it three-dimensional so what would be the volume of that sliver the volume of that sliver will be this function of Y times dy or this whole thing times dy so it would be the integral from zero to B of f of X Y DX that gives us the area of this blue sheet now if I multiply this whole thing times dy I get this volume right it gets some depth that this little area that I'm shading right here gives death of depth of that sheet now if I added all of those those sheets that now have depth if I if I took the infinite sum so if I took the integral of this from my lower Y bound from 0 to my upper Y bound a then at least you know based on our intuition here maybe I will have figured out the volume under this surface but anyway I didn't want to confuse you but that's the intuition of what we're going to do and I think you're going to find out that actually calculating the volumes are are pretty straightforward especially when you have fixed X&Y bounds and that's what we're going to do in the next video see you soon