# Area under hyperbola

by sara_87
Tags: hyperbola
 P: 774 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: $$\int_C[Pdx+Qdy]=\int\int(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dxdy$$ 3. The attempt at a solution We can write the area of the domain as: area=$$\frac{1}{2}\int(xdy-ydx)$$ I know what the graph looks like and i know the parametrisation: x=2cosht y=bsinht but i am to use: area=$$\frac{1}{2}\int(xdy-ydx)$$ what would be the limits of integration?
 Sci Advisor HW Helper Thanks P: 25,251 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.
 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?
HW Helper
Thanks
P: 25,251
Area under hyperbola

 Quote by sara_87 thanks for the limits, i agree. When i parametrize the linear part at the boundary x=3, how does this effect the integrand?
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.
 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) ?