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let's attempt another surface integral and I've changed the notation a little bit instead of writing the surface as a capital Sigma I've written it as a capital S instead of writing D lowercase Sigma I wrote D uppercase s but this is still a surface integral of the function y and the surface we care about is X plus y squared minus Z is equal to zero X between 0 and 1 y between 0 and 2 now this one might be a little bit more straightforward than the last one we did or at least I hope it's a little bit more straightforward because because we can explicitly define Z in terms of x and y and actually we could even explicitly define X in terms of Y and Z but I'll do it the other way it's a little bit easier for me to visualize so if you add Z to both sides of this equation right over here you get X plus y squared is equal to Z or Z is equal to X plus y squared and this is actually pretty straightforward to Vitt this surface is pretty easy to visualize where we can give our best attempt at visualizing it so if that is that is our z axis and that is our x axis and that is the y axis we care about the region X between 0 and 1 so maybe this is x equals 1 and y between 0 and 2 so let's say this is 1 this is 2 and the y area so this is we essentially care about the surface over this over this region of the xy-plane and then we can think about what the surface actually looks like this isn't the surface this is just kind of the range of X's and Y's that we actually care about and so let's think about the surface when x and y are 0 Z is 0 so we're gonna be sitting let me do this in a do it in green z is going to be right over there and now as Y increases as or if we if when X is equal to 0 for just talk about the zy plane Z is going to be equal to Y squared so Z is going to be equal to Y squared so this might be Z is equal to 4 this is Z is equal to 2 1 3 so Z is going to do something like this it's going to be look it's going to be a parabola and the zy plane it's gonna look something something like that now when Y is equal to 0 Z is just equal to X so as X goes to 1 Z will also go to 1 so Z will go like this the scales of the axes aren't they're not drawn to scale the Z is a little bit more compressed than the X or Y the way I've drawn them and then from this point right over here you add the Y squared and so you get something that looks something that looks so this is this is this point there and then at this point when Y is equal to 2 and X is equal to 1 if Z is equal to 5 it's gonna look something like this something like this and then you're going to have the straight line like that at this point is right over there and this surface is the surface that we are going to take or the surface over which we're going to we're going to we're going to evaluate the surface integral of the function y and so one way you could think about it Y could be maybe the mass density of this surface and so when you when you multiply Y times HDS you're essentially figuring out the mass of that little chunk and and and then you're figuring out the mass of this entire surface and so one way you can imagine as we go more and more in that direction is Y is increasing this thing is getting more and more dense so this part this part of the surface is more dense is more dense than as Y becomes lower and lower and then that would actually give us the mass with that out of the way let's actually evaluate it and so as you know the first step is to figure out a parameterization and it should be pretty straightforward because we can write Z explicitly in terms of X and Y and so we can actually use x and y as the actual parameters or if we want to just substitute it with different parameters we could but let me so let's just write let me do that so let me just write X is equal to X is equal to and in the spirit of using different notation instead of using s and T I'll use U and V X is equal to u let's say Y is equal to V and then Z is going to be equal to U plus V squared and so our surface written as a as a vector position function or position vector function our surface we can write it as our which is going to be a function of U and V and it's going to be equal to u i+ v + VJ + u + V squared u + V squared K and then U is going to be between 0 & 1 because X is just equal to U or U is equal to X so U is going to be equal to is going to be between 0 and 1 and then V is going to be V is going to be between 0 and 2 I'm going to leave you there in the next video we'll actually set up the surface integral now that we have the parameterization done