A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

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 \(\oint_C(2x-y)dx +(x+3y)dy \)

  • This Question is Closed
  1. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @dan815

  2. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    you don't wanto use green's thm ?

  3. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    If it helps, why not?

  4. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    use it then

  5. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  6. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    that just means the curve is a closed loop

  7. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  8. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  9. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    knw how to find the curl ?

  10. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  12. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  13. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1434636122182:dw|

  14. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    ok, give me your way, please. hehehe..

  15. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    find the curl first

  16. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    \(\large Mdx + Ndy\) curl = \(N_x - M_y\)

  17. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  18. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  19. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  20. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    I DO lost. :)

  21. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Easy.. just take partials and subtract

  22. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  23. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    because we want to use green's thm

  24. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  26. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  27. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  28. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    oh, that is perimeter of the circle?

  29. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  30. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  31. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    One more question:

  32. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  33. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  34. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  35. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    for all other cases you need to work it by parameterizing

  36. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    YES.

  37. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  39. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  40. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    that red part is 2pi final answer is 4pi

  41. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    how?

  42. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    area of unit circle is pi

  43. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Oops! you're right haha

  44. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  45. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  46. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Yes, Sir.

  47. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.