Area Under Hyperbola: Green's Theorem

[sara_87]

Homework Statement

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

Homework Equations

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

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?

 

[Dick]

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.

 

[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?

 

[Dick]

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.

 

[sara_87]

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)
?

 

[Dick]

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)
?

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