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Studying for a test? Prepare with these 13 lessons on Integrating multivariable functions.
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We're now in the home stretch. We just need to evaluate this third surface integral, which is over in this top part of our little chopped cylinder over here. So let's try to think of a paramaterization- and let me just copy and paste this entire drawing, just so that I can use it down below as I parameterize it. So let me copy it, copy... ...and then go all the way down here, and let me paste it. Okay, that is our shape again, our surface, and then we go to the layer that I want to get on and then let me start. Let me start evaluating it. So what we want to care about is the integral over surface three of z, ds. In surface 3 here, we see that the x and y values essentially take on all the x and y values inside of the unit circle, including the boundary, and that the z values are going to be a function of the x values. We know that this plane, that this top surface right over here, s3, is a subset of the plane z, z is equal to 1-x. It's a subset that's kind of above the unit circle in the x-y plane. or kind of the subset that intersects with our cylinder and kind of chops it. So let's think about x and y first. So first, so x- let's think about it in terms of polar coordinates because that's probably the easiest way to think about it. I'm going to re-draw kind of a top view, so I'm going to re-draw top view so that that is my y-axis, and this is my x-axis, and the x's and y's can take on any value. They essentially have to fill the unit circle. So if you, if you were to kind of project this top surface down onto the x-y plane, you would get this Rn surface, that bottom surface, which looked like this. It was essentially the unit circle, just like that. I'm going to draw it a little bit neater than that, I can do a better job, so that'll be...all right. So let me draw the unit circle as neatly as I can, so there's my unit circle. And so we can have one parameter that essentially says how far around the unit circle we're going, so essentially that would be our angle, and let's use theta, because that's, what, just for fun, because we haven't used theta as our parameter yet. That's theta, but if we had x's or y's, it's just a function of theta and we had a fixed radius, that would essentially just give us the points on the outside of the unit circle. But we need to be able to have all of the x-y's that are on the outside AND the inside of the unit circle. So we actually have to have 2 parameters. We need to not only vary this angle, but we also need to vary the radius. So we would want to trace out the outside of that unit circle, and maybe we would want to shorten it at little bit, and then trace it out again. And then shorten it some more, and then trace it out again. And so you want to actually have a variable radius as well, and so you could have how far out you're going. You could call that r. So, for example, if r is fixed And you change the ranges of theta then you would essentially get all of those points right over there. You would do that for all of the r's, and from r to zero, all the way to r1, and you would essentially fill up the entire unit circle. And so let's do that. So r is going to go between 0 and 1, r is going to go between 0 and 1, and our theta is going to all the way around. So our theta is going to go between 0 and 2π. This is- let me write this down; I wrote 0 instead of theta. Our theta is going to be greater than or equal to 0, less than or equal to 2π, and now we're ready to parameterize it.