1 99:59:59,999 --> 99:59:59,999 We're now in the home stretch. We just need to evaluate 2 99:59:59,999 --> 99:59:59,999 this third surface integral, which is over in this top part 3 99:59:59,999 --> 99:59:59,999 of our little chopped cylinder over here. So let's try to think 4 99:59:59,999 --> 99:59:59,999 of a paramaterization- and let me just copy and paste this entire 5 99:59:59,999 --> 99:59:59,999 drawing, just so that I can use it down below 6 99:59:59,999 --> 99:59:59,999 as I parameterize it. So let me copy it, copy... 7 99:59:59,999 --> 99:59:59,999 ...and then go all the way down here, and let me paste it. 8 99:59:59,999 --> 99:59:59,999 Okay, that is our shape again, our surface, 9 99:59:59,999 --> 99:59:59,999 and then we go to the layer that I want to get on 10 99:59:59,999 --> 99:59:59,999 and then let me start. Let me start evaluating it. 11 99:59:59,999 --> 99:59:59,999 So what we want to care about is the integral over surface three 12 99:59:59,999 --> 99:59:59,999 of z, ds. 13 99:59:59,999 --> 99:59:59,999 In surface 3 here, we see that the x and y values essentially 14 99:59:59,999 --> 99:59:59,999 take on all the x and y values inside of the unit circle, 15 99:59:59,999 --> 99:59:59,999 including the boundary, 16 99:59:59,999 --> 99:59:59,999 and that the z values are going to be a function of the x values. 17 99:59:59,999 --> 99:59:59,999 We know that this plane, that this top surface right over here, s3, 18 99:59:59,999 --> 99:59:59,999 is a subset of the plane z, z is equal to 1-x. 19 99:59:59,999 --> 99:59:59,999 It's a subset that's kind of above the unit circle in the x-y plane. 20 99:59:59,999 --> 99:59:59,999 or kind of the subset that intersects with our cylinder 21 99:59:59,999 --> 99:59:59,999 and kind of chops it. 22 99:59:59,999 --> 99:59:59,999 So let's think about x and y first. 23 99:59:59,999 --> 99:59:59,999 So first, so x- let's think about it in terms of polar coordinates 24 99:59:59,999 --> 99:59:59,999 because that's probably the easiest way to think about it. 25 99:59:59,999 --> 99:59:59,999 I'm going to re-draw kind of a top view, 26 99:59:59,999 --> 99:59:59,999 so I'm going to re-draw top view so that 27 99:59:59,999 --> 99:59:59,999 that is my y-axis, 28 99:59:59,999 --> 99:59:59,999 and this is my x-axis, 29 99:59:59,999 --> 99:59:59,999 and the x's and y's can take on any value. 30 99:59:59,999 --> 99:59:59,999 They essentially have to fill the unit circle. 31 99:59:59,999 --> 99:59:59,999 So if you, if you were to kind of project 32 99:59:59,999 --> 99:59:59,999 this top surface down onto the x-y plane, 33 99:59:59,999 --> 99:59:59,999 you would get this Rn surface, that bottom surface, 34 99:59:59,999 --> 99:59:59,999 which looked like this. 35 99:59:59,999 --> 99:59:59,999 It was essentially the unit circle, 36 99:59:59,999 --> 99:59:59,999 just like that. 37 99:59:59,999 --> 99:59:59,999 I'm going to draw it a little bit neater than that, 38 99:59:59,999 --> 99:59:59,999 I can do a better job, so that'll be...all right. 39 99:59:59,999 --> 99:59:59,999 So let me draw the unit circle as neatly as I can, 40 99:59:59,999 --> 99:59:59,999 so there's my unit circle. 41 99:59:59,999 --> 99:59:59,999 And so we can have one parameter 42 99:59:59,999 --> 99:59:59,999 that essentially says how far around 43 99:59:59,999 --> 99:59:59,999 the unit circle we're going, 44 99:59:59,999 --> 99:59:59,999 so essentially that would be our angle, 45 99:59:59,999 --> 99:59:59,999 and let's use theta, 46 99:59:59,999 --> 99:59:59,999 because that's, what, just for fun, 47 99:59:59,999 --> 99:59:59,999 because we haven't used theta 48 99:59:59,999 --> 99:59:59,999 as our parameter yet. 49 99:59:59,999 --> 99:59:59,999 That's theta, but if we had x's or y's, 50 99:59:59,999 --> 99:59:59,999 it's just a function of theta and we had a fixed radius, 51 99:59:59,999 --> 99:59:59,999 that would essentially just give us the points 52 99:59:59,999 --> 99:59:59,999 on the outside of the unit circle. 53 99:59:59,999 --> 99:59:59,999 But we need to be able to have all of the 54 99:59:59,999 --> 99:59:59,999 x-y's that are on the outside AND the inside of the unit circle. 55 99:59:59,999 --> 99:59:59,999 So we actually have to have 2 parameters. 56 99:59:59,999 --> 99:59:59,999 We need to not only vary this angle, 57 99:59:59,999 --> 99:59:59,999 but we also need to vary the radius. 58 99:59:59,999 --> 99:59:59,999 So we would want to trace out the outside of that unit circle, 59 99:59:59,999 --> 99:59:59,999 and maybe we would want to shorten it at little bit, 60 99:59:59,999 --> 99:59:59,999 and then trace it out again. 61 99:59:59,999 --> 99:59:59,999 And then shorten it some more, and then trace it out again. 62 99:59:59,999 --> 99:59:59,999 And so you want to actually have a variable radius as well, 63 99:59:59,999 --> 99:59:59,999 and so you could have how far out you're going. 64 99:59:59,999 --> 99:59:59,999 You could call that r. 65 99:59:59,999 --> 99:59:59,999 So, for example, if r is fixed 66 99:59:59,999 --> 99:59:59,999 And you change the ranges of theta 67 99:59:59,999 --> 99:59:59,999 then you would essentially get all of those points 68 99:59:59,999 --> 99:59:59,999 right over there. 69 99:59:59,999 --> 99:59:59,999 You would do that for all of the r's, 70 99:59:59,999 --> 99:59:59,999 and from r to zero, all the way to r1, 71 99:59:59,999 --> 99:59:59,999 and you would essentially fill up the entire unit circle. 72 99:59:59,999 --> 99:59:59,999 And so let's do that. 73 99:59:59,999 --> 99:59:59,999 So r is going to go between 0 and 1, 74 99:59:59,999 --> 99:59:59,999 r is going to go between 0 and 1, 75 99:59:59,999 --> 99:59:59,999 and our theta is going to all the way around. 76 99:59:59,999 --> 99:59:59,999 So our theta is going to go between 0 and 2π. 77 99:59:59,999 --> 99:59:59,999 This is- let me write this down; 78 99:59:59,999 --> 99:59:59,999 I wrote 0 instead of theta. 79 99:59:59,999 --> 99:59:59,999 Our theta is going to be greater than or equal to 0, 80 99:59:59,999 --> 99:59:59,999 less than or equal to 2π, 81 99:59:59,999 --> 99:59:59,999 and now we're ready to parameterize it.