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Gauss-Green Cardioid problem not coming out right

  1. Jul 15, 2012 #1
    1. The problem statement, all variables and given/known data
    Here is the region R plotted in the xy plane by the functions
    X(t)=cos(t)(1-cos(t))
    Y(t)=sin(t)(1-cos(t))

    go with
    f(x,y)=3+y-x and g(x,y)=3-x

    calculate ∫∫_R f(x,y) dxdy and ∫∫_R g(x,y) dxdy


    2. Relevant equations



    3. The attempt at a solution
    I know I need to use the Gauss-Green formula so

    X(t)=cos(t)(1-cos(t)
    X'(t)=-sin(t)(-cos(t)+1)+cos(t)sin(t)

    Y(t)=sin(t)(1-cos(t)
    Y'(t)=cos(t)(-cos(t)+1)+sin(t)^2

    a=0
    b=2∏

    f(x,y)=3+y-x

    m(x,y)=0
    n(x,y)=∫f(s,y)ds=-.5x^2+3x+xy

    ∫(m(X(t),Y(t))X'(t)+n(X(t),Y(t))Y'(t)) dt from a to b

    Now I know this is a cardioid with a area of 3∏/2 but when I put everything in I get 18.06. Is my a and b wrong because I swear I have done everything else right
     
    Last edited by a moderator: Jul 15, 2012
  2. jcsd
  3. Jul 15, 2012 #2

    vela

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    The integral won't come out to be equal to the area of the cardioid. Why do you think it does? Your answer of 18.06 is correct.
     
  4. Jul 15, 2012 #3
    lol wow I feel stupid, its measuring a 3d space formed by the cardiod and the plane of f(x,y) and g(x,y). I get it now
     
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