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00:59

Andy S.

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

00:51

Heather Z.

0:00

Simon E.

Virendrasingh D.

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So this is my favorite application of Green's theorem, and it's so classic. So you have in the lips. And now I'm sure you know the equation for the area of a circle. But, you know, maybe you have this memorized or not. But a lot of people don't really know the area of an ellipse. So recalled in an Ellipse looks something like this just a stretch out circle. And now, based on this equation, why is going to be going from be to minus being like that and then X will be going from minus a t A. So you can think about and the lips is just a circle that has a different radius in the X direction than in the Y direction. And so it it actually sort of makes sense the equation. So when a equals B, you just get pi r squared because the radio are the same. But, you know, if the if the X radius is a little bit longer, it stretched out on the extraction. Well, then you just multiply by. You know, however far you go away from the center and the X and then however far away you go in the center in the UAE. So I mean, it's a very logical kind of generalization of the area of a circle. But let's see where it comes from, Okay, so of course A and B are positive numbers. That goes without saying, but let's just say it doesn't hurt to say it, Okay, so we'll be positive numbers. Okay. And here's what we're going to dio Well, the area we noticed the double integral. So here is my region are inside here. But I also know that I have this boundary curve seat. Okay, so the area of our we actually have a way to relate the area of our to a line integral along the boundary curves C. So it's equal to one half the line integral of the curve. And then remember this really special Dr Field minus y X and that's the vector field we're going to integrate. Okay, so that's what the area I'm an ellipsis. But of course I'm not done. I need to actually parameter rise. The curve super called that it's not so hard to parameter rise in a lips. It's a very similar to the parameter ization of a circle, but I just want to specify my my ex radius, if you will. So a cosign teeth and then my y radius be scientific and here t will go just like for a circle between zero and two pi. Okay. And we'll need the derivative to set up our line in a girl. So this will be minus a sign t and then be does nt All right, so we're ready to plug everything into the area is just this line integral. So one half and then we're integrating from 0 to 2 pi of what? Okay, so we have minus Why? So it's minus b sine t dot with ex prime Asti here. So that's minus b sine t and then dot with minus a sign t So that's be times a sine squared t. And then we have X, which is a cosign t dot with while the derivative of Why so why prime a t, which is be co sign t. So we get plus be a co sign square t. Okay, But look how cool this is. So this is the integral from 0 to 2 pi sine squared. Plus cosine squared is one. So what are we left with be a or a times B right? But then we're done. This is just one half times two pi times eight times be which we can rear. So the one half cancels with the to what are we left with pie A be as the area of an ellipse. That's the easiest way I know toe drive the equation for the area of an ellipse But of course it it requires, you know, sort of Ah, pretty deep fear, Um, in calculus screen serum. But again, I mean, you're just seeing how cool green serum is what is able to dio. Not only are you able to take difficult line intervals and convert them into simpler area integral, you're actually able to just express the area of regions by line integral. So it's just super cool. Hope you're appreciating it.

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