Page 1 of 2 12 LastLast
Results 1 to 15 of 19

Math Help - Integral of line

  1. #1
    Super Member
    Joined
    Jun 2008
    Posts
    829

    Integral of line

    Calculate the integral of line \int_c F.dr, where F(x,y) = \sqrt{x^2+y^2} and c is the circumference x^2+y^2=1

    How do I calculate integral of line ?

    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,347
    Thanks
    30
    Quote Originally Posted by Apprentice123 View Post
    Calculate the integral of line \int_c F.dr, where F(x,y) = \sqrt{x^2+y^2} and c is the circumference x^2+y^2=1





    How do I calculate integral of line ?



    Are you sure it's F\cdot r with your given F?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Jun 2008
    Posts
    829
    Yes the answer is 2 \pi
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Calculus26's Avatar
    Joined
    Mar 2009
    From
    Florida
    Posts
    1,271
    Apprentice

    In English we call it a line integral

    The question only makes sense If F is a vector field

    Do you mean F = r = xi +yj ?

    In which case use x = cos(t) y = sin(t) to parameterize the curve

    r = cos(t) i +sin(t) j dr/dt = -sin(t) i +cos(t) j

    On the is curve F = cos(t) i +sin(t) j

    F*dr/dt = 0

    Therefor the line integral is 0


    If you want check out the line integral page on my website for the general method

    Line Integrals
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,347
    Thanks
    30
    Quote Originally Posted by Apprentice123 View Post
    Yes the answer is 2 \pi
    But {\bf F} \cdot d {\bf r} only makes sense if both are vectors!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Calculus26's Avatar
    Joined
    Mar 2009
    From
    Florida
    Posts
    1,271
    dr is understood to be the vector dx i +dy j

    Recall For line integrals to actually calculate we use

    integral of F(x(t),y(t)) *dr/dt as t varies from a to b

    F*dr is just notation rarely used just like the notation integral(fdx+gdy)

    where F = f i + g j

    We use integral of F(x(t),y(t)) *dr/dt
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member
    Joined
    Jun 2008
    Posts
    829
    This is correct ?

    \int_0^{2 \pi} \sqrt{(cos(t))^2 + (sin(t))^2}.(-sin(t),cos(t))dt
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor Calculus26's Avatar
    Joined
    Mar 2009
    From
    Florida
    Posts
    1,271
    what you have makes no sense --F must be a vector field!!!!

    So what is F ?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Super Member
    Joined
    Jun 2008
    Posts
    829
    Sorry, what are the steps to resolve line integral ?
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,347
    Thanks
    30
    Is it possible that what you want is \int_c F\,dx, \int_c F\,dy or \int_c F\,ds?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor Calculus26's Avatar
    Joined
    Mar 2009
    From
    Florida
    Posts
    1,271
    You start with a vector field F = f(x,y) i + g(x,y) j

    parameterize the curve r = x(t) i +y(t) j

    Then then the line integral is the integral of F(x(t),y(t))* dr/dt Where * is the dot product

    Integrate F(x(t),y(t))* dr/dt from a to b
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor Calculus26's Avatar
    Joined
    Mar 2009
    From
    Florida
    Posts
    1,271
    Again check out

    Line Integrals for the development of line integrals
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Super Member
    Joined
    Jun 2008
    Posts
    829
    This line integral is:

    \int_0^{2 \pi} \sqrt{cos(t)^2}(-sen(t))dt + \int_0^{2 \pi} \sqrt{sen(t)^2}(cos(t))dt

    = \frac{-1}{2} \frac{cos(t)^3}{3}|_0^{2 \pi} + \frac{-1}{2} \frac{sen(t)^3}{3} |_0^{2 \pi} = 0

    ????
    Follow Math Help Forum on Facebook and Google+

  14. #14
    MHF Contributor Calculus26's Avatar
    Joined
    Mar 2009
    From
    Florida
    Posts
    1,271
    Apprentice

    What is the vector field F you start with? ---If we don't know F we can't proceed!!!!!!!!!!!!!!!

    F= (x^2 +y^2)^(1/2) is not a vector field

    Look at the problem again there must be a vector field F !!!!!!!!!!!!!!
    Follow Math Help Forum on Facebook and Google+

  15. #15
    Super Member
    Joined
    Jun 2008
    Posts
    829
    This problem i find in internet but not correct.

    This problem is the book:

    1) C is the curve represented by the equations:
    x=2t; y=3t^2; (0 \leq t \leq 1)

    Calculate the line integral along C

    a) \int_c (x-y)ds
    b) \int_c (x-y)dx
    c) \int_c (x-y)dy

    Follow Math Help Forum on Facebook and Google+

Page 1 of 2 12 LastLast

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: December 11th 2011, 11:30 PM
  2. [SOLVED] Line Integral along a straight line.
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 11th 2011, 07:18 PM
  3. Replies: 0
    Last Post: May 9th 2010, 01:52 PM
  4. [SOLVED] Line integral, Cauchy's integral formula
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: September 16th 2009, 11:50 AM
  5. [SOLVED] Line integral, Cauchy's integral formula
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 15th 2009, 01:28 PM

Search Tags


/mathhelpforum @mathhelpforum