We were asked to calculate the flux of the given surface and we're given that Z is in forms of G of x y, merely that Z is equal to X and y eso we will be have to use the equation that thes surface of our flux integral is equal to negative p times the partial with respect to X minus. Q. Times The partial with respect to why, plus our d a, um, plan. In this case, F is going to be equal to our equation, so F is equal to p I plus q J plus Archaic and we're given that f is equal two x y i plus y z j plus z x k. So since we're also given that Z is equal to G of X, why, which is equal to four minus X squared minus y squared? Uh, we're starting to plug into our formula. We actually already have everything we need besides the X partial and the white partial What the X partial of C is going to be d g D acts which is equal to negative X squared drive, which is negative two x and it will be the same for wise, so D g d y is equal to negative. To what? So now we have both the partials we have P Q and R because P Q R are the terms attached i, J and K And that is enough to plug in for integral. Um, because we're also given the X goes from 0 to 1 and why goes from 0 to 1? So playing into our internal, we'll have the integral from 0 to 1 into girlfriends. Yodo One of negative X y cause that's a negative p multiplied by negative two x minus Boise I won't buy by negative two y plus z x d a. And this will be equal to go from 0 to 1 and a girl from 0 to 1 of two x squared y plus two y squared because he plus z x d y d x In this particular problem, the order really won't change anything. Dx dy Why would also work? Justus? Well, as do I. D x and now substituting out Z because we know ZZ Goto four minus expert minus y. We need to get rid of the Z in order to make this a integral in terms of all the X and y. So doing so well give the integral from 0 to 1 Girlfriends, you know, one of two x squared y plus two Why square multiplied by four minus X squared minus y squared plus tax multiplied by four minus X squared minus y squared of the white checks. And this will be equal to the same integral 0101 of two x squared wind plus eight y squared minus two X squared y square. This isn't going to be extremely pretty, by the way. This is 23 term foils when it's two x squared y squared minus two y to the fourth plus four x minus. X cubes X cubed minus X y square, D Y d backs And now we're good to do the first. Integral with respect to what? And this will be equal to the angle. From 01 of X square. Why square plus 8/3 Swyche Cute minus two X squared. Why cubed over three minus 2/5 wide of fifth plus four x wine minus X cubed Why minus X Lies cubed over three d x, and that will be from 01 so sentence firms Jodl one. We're just going to rewrite this both every Why changed to a one just rather nice for us. This will be equal to the integral from 0 to 1 of X squared, plus 8/3 tu minus two X squared over three minus 2/5 plus four x minus X cubes. So there should be a why on this term my four x cubed y, um minus X over three decks. And this is a early integral, just with many different X terms in it's all of these are give me just ah, integrated using the normal, um, rule of integration. So doing this will give the integral from first let's simplify and go from 0 to 1 34/15 plus X squared over three plus 11 x over three minus X cubed d x, and this will be equal. Teoh 34 x over 15 plus X cubed over nine plus 11 x squared over six minus X to the fourth over four from 0 to 1. So again we'll just be swapping out all the exits with ones. So this would be 34 15th plus 1/9 plus 11 6th minus 1/4. From here, we have to find a common denominator. So we need a denominator that has, um if you ever need to find a nominators on, like, quickly, One way to do it is if you just factor them into prime. So in this case, five and three, three and three, three and two. And to and to, um, as long as you make sure each prime is accounted for Ah, you confined your denominator quickly. Highway. So in this case, five times, three times, three times, two times two. Because the most we have is two twos in the four term and two threes in the nine term than a five on the 15. This will give 15. 45 91 80 Denominator is 1 80 I'm from there. Um, I will make the 34 turned to a 408 The one turned into a 20. The 11 will turn into a 330 and the last one will turn into a 45. And then this will give a answer of our flux to be 713 over 180. And this problem was more tedious. than complex right? It was a lot of computation. That's also part of the reason why our answer is so gross. But this would be how you calculate the flux for this particular surface.