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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

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    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.
     
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