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Campbell University

Oregon State University

Harvey Mudd College

Baylor University

00:51

Heather Z.

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

01:02

Anshu R.

01:34

Scott N.

0:00

Jsdfio K.

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Okay, so this is a very similar question. But instead of finding the circulation integral, we want to find the flux integral. And so we have this square, and we could actually just break up the square into its four sides. So here here, hear, hear. And it has a counterclockwise orientation. So it looks like this and what we could find is we could find the flux across each one of the boundaries and add them up. But we would have to do four in a girls, and that sounds like a big pain. So it's gonna be better to just convert it into a single double, integral over this region. And what's even better is this region. So we have the curve c. You have the region enclosed, which we call our the region is very easy to describe. So what does green serum tells us? It tells us that the flock center girl so adding up along the boundary, all the components of the vector field that Z going outside the boundary or the component in the normal direction is equal to the double integral over the enclosed region of the divergence. Basically, how much the field is spreading around each point. So that's gonna be P X plus Q. Why? Where again? This is going to be P. This is going to be keep what is p X? It's gonna be cosign squared. Why? And then what is Q? Why that's gonna be sine squared x. Okay, so then what is the sequel to? It's equal to the double integral over this region of cosine squared. Why plus sine squared x d a. Okay, And so x and y both just go from 1 to 2. So we have 1 to 2 in a girl of 1 to 2 of what Now, let's see. So what can we do here? So, unfortunately, you might be tempted to say, Oh, go sine squared. Why? Plus sine squared X is one. But remember, that's only true if the arguments are the same and they're not here, So we're actually going toe have to evaluate these intervals. So you have cosign squared why plus sine squared X and still just be dx dy y It's not gonna matter which order I integrate sensitive a rectangle. Okay, so it's the anti derivative of sine squared of X with respect to X. Well, here's we're gonna want to use our formulas, our trig identities. This is going to be one minus co signed two X all divided by two and co sign squared is gonna be Well, this is really co sine squared. Why is gonna be one plus cosine to why all over to Okay, Okay, so then what's an anti derivative of sine squared X? I just need to take an anti derivative of this. So we have X and then this is What is this going to be? This is just going to be minus co sign two x over two. But I could just write that as over four and put it to their So this is the anti derivative about that in the anti derivative of this guy is going to be to why it's a very similar co sign to why, over four and then I just want to evaluate this anti derivative from 1 to 2. So that's going to be This is just going to be one when I evaluate X. So it's going to be to and then minus so I'll have co sign of four and then minus minus of plus co sign of too so minus Kosan. Four plus because Sinus to all over four and then plus well, same thing here. I'll just have it, too, when I evaluate why from 1 to 2 and now have plus co sign of four minus co. Sign of to all over for okay and so what happens here? Well, notice that these have opposite signs, so it looks like the coastline. Fours will cancel. The coastline twos will cancel. We'll have to over four plus 2/4. So it looks like our final answer will be one just like that. And I prefer doing that then, actually breaking this up into four line intervals in doing it, having two parameter rise the curves. This is a walk in the park, so green serum, as you can see, is very powerful. It's very useful

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