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Problem 9 Easy Difficulty

Use Green's Theorem to evaluate the line integral along the given positively oriented curve.
$\int_{c} y^{3} d x-x^{3} d y, \quad C$ is the circle $x^{2}+y^{2}=4$

Answer

$-24 \pi$

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Video Transcript

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.

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