Area under hyperbola

  1. Nov 8, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the area enclosed by the hyperbola: 25x^2-4y^2=100 and the line x=3
    using the green's theorem

    2. Relevant equations

    Green's theorem:
    [tex]\int_C[Pdx+Qdy]=\int\int(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dxdy[/tex]

    3. The attempt at a solution

    We can write the area of the domain as:
    area=[tex]\frac{1}{2}\int(xdy-ydx)[/tex]
    I know what the graph looks like and i know the parametrisation:
    x=2cosht
    y=bsinht
    but i am to use: area=[tex]\frac{1}{2}\int(xdy-ydx)[/tex] what would be the limits of integration?
     
  2. jcsd
  3. Nov 8, 2009 #2

    Dick

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    The t limits for the hyperbolic segment of the parametrization are where x=3, i.e. 3=2*cosh(t), yes? Don't forget you need a separate parametrization for the linear part of the boundary x=3 and don't forget to choose a consistent orientation for the two line integrals.
     
  4. Nov 9, 2009 #3
    thanks for the limits, i agree.
    When i parametrize the linear part at the boundary x=3, how does this effect the integrand?
     
  5. Nov 9, 2009 #4

    Dick

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    The integrand is completely different. To do the line part you need to write an x(t) and y(t) that parametrize the line x=3.
     
  6. Nov 9, 2009 #5
    Oh right i see. so when i do that, when i find x(t) and y(t) for the line, and the X(t) and Y(t) for the hyperbola part, how do i out this in the integrand?
    I mean for the xdy part, is this: (x(t)+X(t))dy(t)
    ?
     
  7. Nov 9, 2009 #6

    Dick

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    Why don't you just do two separate line integrals instead of trying to mix them up? That's what I would do.
     
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