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# Triple integrals 3

## Video transcript

let's now do another triple integral and in this one I won't actually evaluate the triple integral but what we'll do is we'll define the triple integral we're going to we're going to do something similar that we did in the second video where we figured out the mass using a density function but what I want to do in this video is show you how to set the boundaries when the figure is a little bit more complicated and if we have time we'll try to do it where we change the order of integration so let's say I have the surface let me just make something up 2x 2x plus 3z plus y is equal to 6 let's draw that surface look something like this this will be my x-axis it's going to be my z-axis that's going to be my y-axis draw them out X Y & Z and I care about the surface in the kind of positive octant right because when you're doing with three dimensions we have instead of four quadrants we have eight octants but we want the octant where all XY and z is positive which is one I drew here so let's see let me draw some what is the x-intercept when y and z are zero so we're right here that's the x-intercept 2x is equal to 6 so X is equal to 3 so 1 2 3 so that's the x-intercept the y-intercept when x and z are 0 or on the y-axis so Y will be equal to 6 so we have 1 2 3 4 5 6 the y-intercept and then finally the Z intercept when x and y are 0 or along the z axis 3 z will be equal to 6 so Z is equal to 1/2 so the way it would let's see so the figure that I care about will look something like this it'll be this inclined surface it will look something like that and this positive octant so this is the surface defined by this function let's say that I care about the volume and I'm going to make it a little bit more complicated we could say oh well that's what's the volume between the surface and the xy-plane but I'm going to make it a little bit more complicated let's say the the volume above this surface and the surface Z is equal to 2 so essentially the surface Z so the volume we care about it's going to look something like this let me see if I can pull off drawing this so if we go up to here so if we go up to here and then we have let me draw at the top in a different color let me draw the top in green so this is along the zy plane and then the other edge is going to look something like this let me make sure I can draw it this is the hardest part so go up to here and then this is along the ZX plane and we'd have another line connecting these two so this green triangle this is part of the plane Z is equal to 2 but what we can do is well we what the volume we care about is the volume between this top green plane and this tilted plane defined by 2x plus 3z plus y is equal to 6 so this area in between let me see if I can make it a little bit clearer because the visualization as I say is often the hardest part so we'd have kind of a front wall here and then the back wall would be this wall back here and then there'd be another wall here and then the base of it the base I'll do in magenta will be this plane so the base is that plane that's the bottom part anyway I don't know if I should have made it that messy because we're gonna have to draw DVS and D volumes on it but anyway let's let's try our best so if we're going to figure out the volume and actually let's since we're doing a triple integral and we want to show that we have to use a triple integral instead of just doing it volume let's do a the mass of something a variable to so let's say the density in this volume that we care about the density function it's a function of XY and Z it can be anything that's not the point of what I'm trying to teach here but I'll just define something let's say it's x squared Y Z our focus here is really just to set up the integrals so the first thing I like to do is I visualize what we're going to do is we're going to set up a little bit a little cube in the volume under consideration so if I had a let me do it in a bold color so that you can see it so if I have a cube maybe I'll do it in brown it's not as bold but it's different enough from the other colors so if I had another I've had a little cube here in the volume under consideration that's a little cube you could consider that DV the volume of that cube is kind of a volume differential and that is equal to DX and outside this is dy let me do this in yellow or green even better so dy which is this like dy times DX DX times DZ right that's the volume of that little Q and if we wanted to know the mass of that cube we would multiply the density function at that point times this DV so the mass you could call it D I don't know DM the mass differential is going to be equal to that times that so x squared Y Z times this dy DX DZ and we normally switch this order around depending on what we're going to integrate with respect to first so we don't get confused so let's try to do this let's try to set up this integral so let's do it traditionally the way the last couple triple integrals we did we integrated with respect to Z first right so let's do that so we're going to integrate with respect to Z first so we're going to take this cube and we're going to sum them up we're going to sum up all of the cubes in the z-axis so going up and down first right so if we do that what is the bottom boundary what is the bottom boundary well the bottom of so when you sum up up and down these cubes are going to turn into columns right so what is the bottom of the column the bottom bound well it's the surface is the surface defined right here so if we want that bottom bound to defined in terms of Z we just have to solve this in terms of T so let's subtract so what do we get if we want this defined in terms of Z we get three Z is equal to 6 minus 2x minus y or Z is equal to 2 minus 2/3 X minus y over 3 this is the same thing as that but when we're talking about Z explicitly defining a Z this is how we get just algebraically manipulated so the bottom boundary and you visualize it right the bottom of these columns can go up and down we're going to add up all the columns in up in down direction right you can imagine summing them the bottom boundary is going to be this surface Z is equal to 2 minus 2/3 X minus y over 3 and then what's the upper bound well the column by the top of the column is going to be this green plane and we what did we say that green plane was well Z is equal to 2 Z is equal to 2 and that's this plane this surface right here Z is equal to 2 and of course what is the volume of that column well we're it's going to be the density function x squared Y Z times the volume differential but we're integrating with respect to Z first let me write DZ there and this I don't know let's say we want to integrate with respect to I don't know we want to integrate with respect to X for next in the last couple of videos I integrated with respect to Y next let's do X just to show you it really doesn't matter so we're going to integrate with respect to X so now we have these columns right when we integrated with respect to Z we get the volume of each of these columns where the top boundary is that plane see if I can draw it decently the top boundary is that plane the bottom boundary is this surface and now we want to integrate with respect to X so we're going to add up all of the D X's so what is the bottom boundary for the x's well this surface is defined for all the way to X this the volume under question is to find all the way until X is equal to zero and if you get confused and it's not that difficult to get confused when you're imagining these three-dimensional things say you know what we've already integrated with respect to Z the two variables I have left or x and y let me draw the projection of our volume onto the XY plane and what does that look like so I will do that because that actually does help simplify things so if we twist it so if you take this Y and flip it out like that X like that we'll get in kind of the traditional we'll get it in the traditional way that we learned when you first learned algebra the XY axis so this is X this is y and this point is what or at this point what is that that's X is equal to 3 so it's 1 2 3 that's X is equal to 3 and this point right here is y is equal to 6 so 1 2 3 4 5 6 and so on the XY axis kind of the domain you could view it that way it looks something like that and so one way to think about is we've figured out if these columns we've integrated up-down or along the z axis but when you view it looking straight down onto it you're looking on the XY plane each of our columns are going to look like this where the base of the column where the where the complem to pop out out of your screen in the z direction but the the base of each column is going to be DX like that and then dy up and down right so we decided to integrate with respect to X next so we're going to add up each of those columns in the x direction in the horizontal direction so the question was what is the bottom boundary what is the lower bound in the X direction well it's X is equal to 0 if there was a line here then it would be that line is we're probably as a function of Y or definitely as a function of Y so our bottom bound here is X is equal to 0 and what's our top bound I realize I'm already pushing the group well our top bound is this relation but it has to be in terms of X right so what is this relation so you could view it as kind of saying well Z is equal to 0 what is this line what is this line right here so Z is equal 0 we have 2x plus y is equal to say we want the relationship in terms of X so we get 2 X is equal to 6 minus y or X is equal to 3 minus y over 2 X is equal to 3 minus y over 2 and then finally we're going to integrate with respect to Y and this is the easy part so we've figured out we've integrated up and down to get a column these are the bases of the column and so we've integrated in the x-direction now we just have to go up and down with respect to y or in the XY plane with respect to y so what is the Y bottom boundary well it's 0 y is equal to 0 and the top boundary is y is equal to 6 and there you have it we have set up the integral and now it's just a matter of chugging through it mechanically but I've run out of time I don't want this video to get rejected so I'll leave you there see you in the next video