## Loser66 one year ago Let C be the circle $$x^2+y^2 =1$$ oriented counterclockwise in the xy-plane. What is the value of the line integral $$\oint_C(2x-y)dx +(x+3y)dy$$

1. Loser66

@dan815

2. ganeshie8

you don't wanto use green's thm ?

3. Loser66

If it helps, why not?

4. ganeshie8

use it then

5. Loser66

ok, let me try. Actually, I didn't know what the notation $$\oint$$ mean. I never see it before. :)

6. ganeshie8

that just means the curve is a closed loop

7. Loser66

I am working on it, will tag you to check it later, ok?

8. ganeshie8

ok, just need to find the curl and setup double integral

9. ganeshie8

knw how to find the curl ?

10. Loser66

yes, I divide it into 2 parts, $$y = \pm \sqrt{1-x^2}$$ ,hence the limit for the first part will go from 0 to 1, right?

11. Loser66

oh, We talk about 2 different things. ha!!

12. ganeshie8

lol yeah actually we don't need to do much work here, find the curl, you will know why :)

13. Loser66

|dw:1434636122182:dw|

14. Loser66

15. ganeshie8

find the curl first

16. ganeshie8

$$\large Mdx + Ndy$$ curl = $$N_x - M_y$$

17. ganeshie8

$$\large (2x-y)dx +(x+3y)dy$$ curl = ?

18. Loser66

It looks like differential equation part? finding exactness, right?

19. ganeshie8

$$\large (2x-y)dx +(x+3y)dy$$ $$M = 2x-y$$ $$N = x+3y$$ $$N_x = 1$$ $$M_y = -1$$ curl = $$N_x - M_y = 1-(-1) = 2$$

20. Loser66

I DO lost. :)

21. ganeshie8

Easy.. just take partials and subtract

22. Loser66

I know, but don't know why we have to do that.

23. ganeshie8

because we want to use green's thm

24. Loser66

I was taught that I have to find parametric equations for x, y and replace and take a loooooooooong steps to get the answer. This is somehow different.

25. Loser66

But that is the reason i post the problem here to learn the shorter way. :)

26. ganeshie8

$\oint_C(2x-y)dx +(x+3y)dy ~~=~~ \iint_R~2 dxdy = 2\iint_R~1 dxdy = ?$

27. Loser66

You still use x, y , not r and theta?

28. Loser66

oh, that is perimeter of the circle?

29. ganeshie8

I just applied green's theorem to convert line integral into double integral

30. Loser66

Ok, I got you. Thanks a lot. Need practice more. :)

31. Loser66

One more question:

32. Loser66

If the curve is not a circle, we must define the limits of x,y to put into the double integral, right?

33. ganeshie8

green's theorem works only if the curve is a closed loop

34. Loser66

Yes, Again, don't we have to change to polar form?

35. ganeshie8

for all other cases you need to work it by parameterizing

36. Loser66

YES.

37. ganeshie8

you can but its not really needed here if you recall the fact that $$\iint_R ~1 ~dxdy$$ represents the area of the region.

38. ganeshie8

$\oint_C(2x-y)dx +(x+3y)dy ~~=~~ \iint_R~2 dxdy = 2\color{red}{\iint_R~1 dxdy }= ?$ that red part represents the area of the circular unit disk

39. Loser66

hey, on the previous comment (and you delete it), you stated the result is 4pi, ha!! now it turns to 2pi??

40. ganeshie8

that red part is 2pi final answer is 4pi

41. Loser66

how?

42. Loser66

area of unit circle is pi

43. ganeshie8

Oops! you're right haha

44. Loser66

hihihi... ok, got you now. Much appreciate for being patient to me.

45. ganeshie8

np :) maybe for practice, work it by parameterizing also

46. Loser66

Yes, Sir.