Area under hyperbola

  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. Dick

    Dick 25,738
    Science Advisor
    Homework Helper

    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. thanks for the limits, i agree.
    When i parametrize the linear part at the boundary x=3, how does this effect the integrand?
     
  5. Dick

    Dick 25,738
    Science Advisor
    Homework Helper

    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. 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. Dick

    Dick 25,738
    Science Advisor
    Homework Helper

    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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