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Use Green's Theorem to evaluatedr. (Check the orientation of the curve before applying the theorem:)F(x , Y) = (y - cos(y), X sin(y)) , C is the circle (x - 4)...

Question

Use Green's Theorem to evaluatedr. (Check the orientation of the curve before applying the theorem:)F(x , Y) = (y - cos(y), X sin(y)) , C is the circle (x - 4)2 + (y + 8)2 = 9 oriented clockwise61

Use Green's Theorem to evaluate dr. (Check the orientation of the curve before applying the theorem:) F(x , Y) = (y - cos(y), X sin(y)) , C is the circle (x - 4)2 + (y + 8)2 = 9 oriented clockwise 61



Answers

Use Green's Theorem to evaluate $ \displaystyle \int_C \textbf{F} \cdot d\textbf{r} $. (Check the orientation of the curve before applying the theorem.)

$ \textbf{F}(x, y) = \langle y - \cos y, x \sin y \rangle $,
$ C $ is the circle $ (x - 3)^2 + (y + 4)^2 = 4 $ oriented clockwise

So in this video, tickle off the falling field, uh, would have been serving challengeable along the curve of the circle Uh uh, over a circle of radius, too. And we're told that the orientations Cho course so a weapon use green steer green serum sense that the line into crows He spoke to the double integral over. Good deal me he fueled by York's finest. If you boys want you so deep you buy the nexus simpering integral of X, Sign y with respect to X just just signed line. And then the people idealizes tentacle Why, minus co sign live with respect to why is just one plus sign And now what we're gonna do is we're gonna take this interval and, uh, polar coordinates. So our d a is just for your data sign online. They could have signed one. It can soak us. That native has been a distribute us for the limits of integration. Again, we said, This is a circle of radius to so far goes from Sorry, it's a circle readies to So our goes from 0 to 2 on the angle data. Since it's a complete, it's a circle goes from 0 to 2. So basically, we're just left with negative one. Are you going straight? Oh, so derivative. So we're gonna pull that money has to be outside. Here we have on the outside of the integral far with respect bars just parts were divided by two and the limits of integration on from zero to a little football. So negative two squared, divided by 2.0 spread divided like do, which is just turn. We can pull that to be outside. So a lot of negative to on the integral off the fate US of data limits of integration are from city through palm So negative 200 outside and 2.9 zeros was to part. So they got a few times two is negative for remember, our integration was clockwise but are coming so or convention as we take the line Integral. We want to be critical for us. So all we have to do switch back minus a positive. We're very of the line into group some 34

Okay. Today I'm going to show you how to use the green Syrian. So in here we need to calculate f er and take the line to go. So in physics, we be say this one is calculated force. Then we can apply for the close surface in I mean for the clothes line Close contour here it's a circle. So when it's a circle so and it is close so we can apply the green Syrian. So then we usedto this one to cross off f and we get Ah so by checking the so orientated the orientation So here is two d So you imagine we have in two D surface, right? And then we take the crows Should be three d point to the up or point to down so should be k So we get minus k here. So should be should be cork arise in this case cork arise. So then we apply the so then we use ceasing here sign And what sign? Why minus one minus sign Why here is the the X sign Why d X and here is the d y. Why minus course and why here. Then we get the I mean hope here. So we get minus K. Sam is here, so the whole thing will become minus dx dy y for the circle. Since this one is the it's account, it's a corporate orientation. We we say this one should be should be so Okay, we talk her here. So we we knew here it's a counterclockwise so and we knew the the early off circle is two pi r r is equal to So it's four pie here and from here we also can use the polar coordinating to integral to this one. I think that way it's easier because we knew we're knew the formula. So So since they have minus here So we need to free because this one is is cork arise. If it is counter cork rights should be minus four pi

A problem. Number four using the serum 17 0.4 point one and green Thierer The integration off F X right D x cross G x Why and b y is equal toe portion re partial X minus person dream parts That Why being Yeah, where d a is equal toe the x de y So we have X squared. Plus y squared is equal to a nine so to the X so so. Two x The X is equal to a negative, so why de y? So b y is equal to negative x over y a b x. So for X squared minus y squared is equal tohave off. That's why and X is equal to g x x y, Then on the integration off X squared minus y squared the X plus x d y is equal to partial X partial X minus Project X squared minus y squared over. Fortunately, why the X? Me? Why? Which is equal to one plus So why the X e y, which is equal to the level integration from negative 3 to 3 negative square root off the nine extra square we rolled off the line X squared or one loss. So why D y the act? But we know that we can use D Y equal to negative X over y the act and why squared is equal to nine minus x square. So, um, so we can say that this is equal to integration from Medicare. Three. The three wide cross wide square from negative square root off mine excess grad toe square root online. X squared the X, which is equal to integration from many to 3 to 3 to produce off nine minus X squared D X, which is equal to 29 over to 5/2 minus nine over two minutes of five over to which is equal to nine point.

Problem. 185 Using green serum, we're gonna set up this integral, okay? And then simplifying This comes out to seven minus three d A, which is to get into four times the area, then finding the area. It's, ah, circle with Radius three. It's a pi r squared and that goes 36 pie.


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