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Area under hyperbolaby sara_87
Tags: hyperbola 
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#1
Nov809, 03:59 PM

P: 774

1. The problem statement, all variables and given/known data
Find the area enclosed by the hyperbola: 25x^24y^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(xdyydx)[/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(xdyydx)[/tex] what would be the limits of integration? 


#2
Nov809, 07:18 PM

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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.



#3
Nov909, 05:02 PM

P: 774

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


#4
Nov909, 06:03 PM

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Area under hyperbola



#5
Nov909, 06:14 PM

P: 774

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) ? 


#6
Nov909, 10:12 PM

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