In order for us to evaluate the surface integral of Stokes, assume just the double integral along the surface of Dell Cross f dotted with the unit normal vector D s. We first must select a viable surface on which to evaluate this. And because we're given in the problem that the boundary coop is X squared plus y squared equals 12. If we select Oh, surface based off of this was Sofus is going to be 12 minus x KWood minus y swallowed. It makes the evaluation of this into role fairly easy. So then all we have to do is start evaluating this interval. So starting with this Del Cross F if we set up a matrix with I J K. And then it's Dell first. So that's the postal with respect to X with respect. A Why, with respect to Z and then which is two why negative z and X. This is equal toe one negative one and negative too, which is found by taking the discriminative of this matrix. So now that we have Dele Cross F, we have to find the unit normal vector, and this is going to be equal to the ingredient of always surface. You calling s divided by the magnitude because it's a unit. A normal vector. The ingredient of a surface here is going to be negative. Two x for the i component negative to why, with the J component and one sweeper K component divided by the magnitude here which is going to be the square root of four x squared plus four y squared plus one. Now, if I frank this so we have more space to maneuver Now we have two dot ah unit normal vector. Just you, with, uh, still cross us. So this leaves us though cross us dotted with the unit Normal vector is going to be equal to weaken. See, it's one times negative two x so that negative two x and then negative one times negative to wise That's plus two Why and negative two times one which is negative two all divided by squirt of four X squared plus for y squared plus one. And before we plug this into the integral, we have ah, Dell cross f dotted with the unit Normal vector in terms of accent. Why over this is a DS and in order for this ds to be in terms of X and y we have to If we say this is ah, central function D s is going to be equal to se. The potion will suspect the X Schoon plus the postal Inspector y squared plus one d A, which is the X t y. So if we take this quote of thes poche ALS and one we see that DS is going to be equal to swell loot of four ex quote, which is the postal with respect to X. It s squared. Plus for why screwed close one, Do you? A. So now if we set up our interval, you see this integral? Well, a leader now is going to be equal to negative two X plus two y minus two All over this quote off four X squared Plus for why squared plus one kinds the square root of four X squared plus four y squared plus one e A. And as you can see the squalid, you will cancel each other out. So if we again shrink this down to get more room to manoeuvre, all we have to do now is evaluate this integral, which I will we wait help you. So we have this double interval of negative two X plus two y minus two d A. And however this area of which integrating is going to be this circle X squared plus y squared is equal to 12. And because we are integrating basically the area of this so call here if we convert toe polar coordinates meaning we put X and y in terms of all in Seda, it will make the limits of these intervals much easier. So if we say X is equal to ah coastline Fada, why is equal toe all times? Sign of Fada and D A, which is equal to D Y D X is going to be equal toe Ah de are de so then if we plug this in, if we first look at a balance weaken, see that all is going to go from zero to the squad of 12 and Seda will be going from 0 to 2 pot. So we have ah, double integral off Negative, too Times X, which is Ah, coastline Seita plus two times why, which is ah sine theta minus two, all multiplied by all D. O D. Seda. And of course, the limits of intervals here, uh, going to be all just from zero toe screwed of 12. And then they'd off with this out of one, which is from 0 to 2 pi. And again. Let me shrink this. So we have more room. Two minutes. It appears oath Ada got caught, but it will related. So now if we evaluate this integral food with, uh you see, it's from Monsieur two pi, and if we distribute this all, you get negative to all squared coast data plus two squared sine theta minus two R D o d theta. Which is of course, equals from 0 to 2 pi of negative 2/3 uh, cubed whose data? Plus 2/3 a cube sine theta minus our squad from zero to the sward of 12 di fada. And again it oppose. We're running out of space. So if we shrink, move this up top. And if we plug in our screwed of 12 0 we're left with the integral from zero. Your two pi of negative 2/3 times 12 12. Coast data plus 2/3 time is 12 Moot. 12 Sign Fada minus 12 de fate up. So now again If we integrate, we have negative 2/3 times 12 boot 12 sine theta because we've integrated plus 2/3 times 12 Route 12 Negative co sign data minus 12. Data from 0 to 2 pi. And so if we look at this first Pote because sign of two pi and sign of Zago of both zero, we can disregard it. And if we look at this second tomb who co sign of two pi is one as his co signing zero. That means we're basically subtracting, uh, the same thing from itself. So that will also go. It is, you know, So this just leaves us with 12 data 0 to 2 pi which is equal to negative 24 pi.