1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Is this correct. Evaluate the line integral

  1. May 30, 2009 #1
    Evaluate the line integral [tex] \int F \circ dr [/tex] for

    (a) F(x,y) = (x - y) * i + xy * j and C is the top half of a circle of radius 2.


    Here's green's theorem. Double integral ( dQ/ dx - dP/dy) dA

    dQ/dx = y
    dP /dy = -1


    It becomes ∫∫ (y +1) dA

    y = r sin Θ




    so then it becomes ∫∫ (r sin Θ + 1) r dr dΘ
    I changed it into cylindrical coordinates


    then 0 < r < 2
    and 0 < Θ < pi


    then we integrate respect to r

    and we get ∫ r^3/3 sin Θ + r^2/ 2 d Θ r is from 0 to 2

    we plug in r

    ∫ 2^3/3 sin Θ + 2^2/ 2 - 0 d Θ = ∫ 4 sin Θ + 2 d Θ

    then we integrate for theta

    and we get
    - 8/3 cos Θ + 2Θ from 0 to pi = - 8/3 cos pi + 2 pi - -8/3 cos 0 + 2(0)


    = -8/3(-1) + 2pi - [ -8/3 -0]



    = 8/3 + 2pi + 8/3 = 16/3 + 2pi
     
    Last edited: May 30, 2009
  2. jcsd
  3. May 31, 2009 #2

    HallsofIvy

    User Avatar
    Science Advisor

    2^3/3= 8/3, not 4!

    Greens theorem equates [itex]\int\int ( dQ/ dx - dP/dy) dA[/itex] to the integral around the closed path forming the boundary of the region. Your original problem " C is the top half of a circle of radius 2" does not have a closed path. You could, after correcting this calculation, find the integral from (-2, 0) back to (2, 0), along the x-axis, and subtract it off.

    Are you required to use Green's theorem? Just a straight path integral does not look difficult.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook