How Does Green's Theorem Simplify Calculating a Line Integral for an Ellipse?

[Calpalned]

Homework Statement

Use Green's Theorem to evaluate the line integral along the given positively oriented curve. $\int_C y^4 dx + 2xy^3 dy$, C is the ellipse $x^2 + 2y^2 = 2$.

Homework Equations

Change of variables: $\int \int_S f(x(u,v),y(u,v)) |{\frac {\partial(x,y)}{\partial (u,v)}}| du dv$

The Attempt at a Solution

How do I change the ellipse to a circle? Is there a way to determine u and v?

 

[STEMucator]

Please refer back to post #3 in your prior thread:

https://www.physicsforums.com/threads/using-greens-theorem.810989/

You could have posted there.

Anyway, after parametrizing the ellipse, you should know what the limits on the integral are for $r$ and $\theta$ without very much thought.

While a change of variables $x = au$ and $y = bv$ would map the ellipse to a circle of radius $\sqrt{2}$, this is unnecessary and a bit of extra work since you still need to calculate the Jacobian.

I suggest just thinking about it for a second.