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Point) Let C be the positively oriented circle 22 + y? =1 Use Green's Theorem to evaluate the line integral Jc lby dx + 5x dy:...

Question

Point) Let C be the positively oriented circle 22 + y? =1 Use Green's Theorem to evaluate the line integral Jc lby dx + 5x dy:

point) Let C be the positively oriented circle 22 + y? =1 Use Green's Theorem to evaluate the line integral Jc lby dx + 5x dy:



Answers

Use Green’s theorem to evaluate line integral $\oint_{C}\left(y-\ln \left(x^{2}+y^{2}\right)\right) d x+\left(2 \arctan \frac{y}{x}\right) d y,$ where $\mathrm{C}$ is the positively oriented circle $(x-2)^{2}+(y-3)^{2}=1.$

So even a disliking to grow the review of this with respect to X Ah, the ripped him off. This with respect to So here is three x square. So the derivative off this component with respect of exes. Syria minus three X square. Mine is the revealed This was respect. Who's ah to the wife Should be, Ah, three y square. Hey, so it should be minus three x square plus y square d A and the circle radio's for Ah, maybe it's even easier if we do a change of coordinate use. Ah, it was a portal crossing the so Oh, ex hos are Earl coz I see that why Khosrow size either and the a host road the road this's and this keeps us us This will gives us ah x square plus y square is throw square And, uh, Roe is from zero to radius is square with forged too And I said, Ah, it's a whole circle So oh, so that should be zero two to pie row Q d ro tc. So here inside the Gogol wrote with a fourth over four and we're probably too uh, Therefore, we are two to the four over for which is for and this is just a concert. So in the root, it's just beautification. Four times Supervise a pie and time stories twenty for pie.

Okay, So this question was asked to determine using greens here on the line to move them faster, determines along the curve, Exclude. Plus my security for so basically just the regular circle. All right, so this is green Steer suffering two screens here. Um, so, first of all, our Q is negative, x cubed. The derivative of that is negative three X squared. And our PC is why cubed and derivative of that is negative. Three. All right, so now, um, or this derivative, what we're gonna do Get up. Eso use polar coordinates to determine it. So, using for court nets, we have rt already stayed on. And if we look at our triangle, we know that our radius Berries from zero to where are angle finger dairies from zero. So this is why so are our various from there too. That's our limit of integration. That's beta various from 0 to 2. All right, now what we notice right here is that we can pull out a negative three is a common factor. So filled it out right here. What? We're left with his X squared plus watch square. Now, this is very interesting, because what do we know what's expert plus y squared Well, X squared plus twice where it is just the first script. So here we have our swear times are told that negative three fold over. So now we have this negative three and the integral off our pure PR. So we're working on the inside trying to work on distant to grow Friday we're here. So the integral off R Q p. R is just far to the power four divided by four. And you could just full of love that one sign on our limits of integration is from zero to. So we put this plug in your on here, so he gets there two times two is four defense for defective. 16 for this right here, 16 which is a constant bullets to the outside. So we have a negative three times 16 divided 54 So 16 diverted by fours for we have to thank you for 12. But once the integral off. So we pulled that 16 out. So we have one rule of one defeat. Well, that's just so we have negative 12 on the outside. Have freedom. Very sperm. 0 to 5 minutes of integration. So Now we get negative 12 times, but we're going to subtract Europe from two ply. So we're left with you 12 times to prime, which is negative. 24. So this until are lining.

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.

I have a problem. 1 88 Using green serum gonna value the Santa Groll. This comes out to double integral of two D A which is unjust, unequal. Ah, Two times a. Then we can find the area individually. Is it gonna be pi times a a squared so total it's two pi squared.


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