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Evaluate the line integral by using Green' $ Theorem(a) Nydr - xdyCis the circle with center the origin and radius 8....

Question

Evaluate the line integral by using Green' $ Theorem(a) Nydr - xdyCis the circle with center the origin and radius 8.

Evaluate the line integral by using Green' $ Theorem (a) Nydr - xdy Cis the circle with center the origin and radius 8.



Answers

Evaluate the line integral by two methods: (a) directly and (b) using Green's Theorem.

$ \displaystyle \oint_C y \, dx - x \, dy $,
$ C $ is the circle with center the origin and radius 4

Given that the equation of a circle is equal to place. The parametric representation is going to be for X. Equal to focus. T why go to 4th 19. So take the derivative to the eggs. It's equal to that. Then the way and swell. It's equal to this. To the limits of integration will be from servitude by because by convention the path must be a rented counterclockwise. Unless specify. So we have ACLU's seeker oversee why the eggs. My let's X. The why so we substitute their limits and the values of X. Y. Why the eggs and eggs. Dy. So this implies that you have the integer from syria to to buy. You have four sign. See my name's four sign. See I was sorry C. D. T. Mine is full course C. Then four cause see D. C. So this then this thing it's equal to from 0 to 2 by we have minus 16 sign squared C. So it's great to see minus 16 hose squared C. D. C. So this day don't forget 16 it's common. So we bring minus 16 out see how several to buy. Science. Great E. Plus course gracie would give us one so we have it's seen. So this is going to be my next 16. You have C. From zero to to buy so minus 16 two pi and this gives us minus 32 minus 32 minus 32. Okay, if you evaluate the integral to directly then for the second part the so for B using Greece Green's theorem, you have dan Segre P. D. A. S less cute the way to be equal to. Then sicker over D. Partial derivative of Q. With respect to eggs. Then mine especially relative of P. With respect to why the X. D. Y way D. So where uh D. S. The disk is a disk X squared plus Y squared less than or equal to 16. So if you find the partial derivative. So this implies that from here special derivative of Q. With respect to X minus. Special derivative of P. With respect to why will be equal to my next one minus one which is minus two. So then our insignia integer. Nobody will be minus two X minus two. The X. The X. The Y. Two constants out. We have been sicker over the X. D. Y. So as a city girl, we have a so we have a two vehicles to the W. C. T. The X. The Y. Which is equal to Hi fourth squared which is 16. Hi. Okay. So this is the area includes by the circle of radios. For so tell it's in place. We have DNC girl. Why the s minus X. Dy two vehicles to negative two times 16. Hi. And this would give us negative 32. Okay, as a finer answer. Which is equal to he's doing the direct method

Hi, Problem. 443 All right, so using green serum, we're gonna set up our Anna grow as okay, and this is gonna be grow from one minus three D A. That comes out to negative 18 pie.

In this video, we're gonna determine the line integral using it's the 1st 1 is gonna be directly, and 2nd 1 is gonna be used. All right, So if we're gonna do it directly, the first thing we're giving is that our curve is that of a circle off radius, too. And then, well, the first thing we're gonna do is we're gonna parameter rise the circle. So t goes from 0 to 2 pi that are X is gonna be to co sign t are wise to send this just regular parameter organization of and then we're gonna determine the X DX is just a derivative off to co scientific perspective. So that their evident Alexis just negative to assign TV that you're a bit of flies just too closely. We are just taking the derivative with respect to X t. Sorry. And now what we're gonna do is we're gonna forget full or effects is analyzed. The X Isn't you wise to this man, solve our integral with changer integral to something, rivers, another limit This from Syria, Jim Pie. And then we're gonna plunge everything rehab before to the original equipment through the original integral. We have So we have to course 90 minus 2 70 times negative to assign plus two course I t plus two ST Times to course, like now if we distribute So for example, strip Yes, but we get is negative for scientific oocyte T plus four Science Square TVT most for coastline square T bus for scientific coastline. Now the usual canceled. So we're just like we could blow to force a comic. So I left with four times, even to quote from 0 to 2 pi science square T plus closeness. But Science square, Teeples Coastlines birthday. So this entire great here just one. So we have to be. So we have There are four times the integral from 0 to 5 DT. The role of the tea is just tea on our limits off integration are from 0 to 2 pi so that we get four times to find my zero which is eight so there since using divided so another way. And this is directly we construct with the line integral directly in another way, we could have determined the lighting to grow is by using green stare of which is right over. So this is screens here So, first of all, our it explains why so incredible? Original equation? Oh, it's attached to the X has exploited spice, for he is exploited by our Cuba's X plus one. Now the derivative of people respect to wise just that negative one because of derivative expert perspective boys. Zero negative. I would respect a wise at the durability off Q. Respected ex. It's just what Now we put this back a door into grow that remember our It's, uh we have t a and that is where we're polar coordinates R D A's Just argue your Sita. So are our ranges from zero to our theater from zero to All right, so now we get the double integral to Lord ER data and now the integral of c R two RTR is just part inspired because we're gonna raise the power are by one. So that's our square that we're gonna be black by. So now we have the to grow from 0 to 2 pi our square as our break is from Syria to defeat. So that's just for so the press. Just 40 painter. No, the integral off work you pay them just for figure as fate over the living off The integral is from Syria to pie. So we click this in two years so we get four times 25 minus zero, which is a heart. So just like we expected, we should get sing answer whether we compute the line integral directly or were computed using the double integral to get the senior

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.


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