Green's Theorem (c = ellipse)

[Melissa00]

Hi

I have to make use of Green's theorem to calculate this:


along the ellipse E: x²+4y²=4

First, I wrote down the parametric equation of the ellipse

Is this even necessary?I don't really know how to make use of this when using Green's theorem.
With the theorem I calculated that I'll need to integrate the double integral of 2.

For the limits of the integrals I thought about this:
Making use of the ellipse equation, I said that x= +/- 2*sqrt(1-y²). Those would be the integration limits for dx Looking at that, for dy= +/- 1?

I'd greatly appreciate your help with this!

Thanks

 

[DrPhil]

Hi

I have to make use of Green's theorem to calculate this:

View attachment 3124
along the ellipse E: x²+4y²=4

First, I wrote down the parametric equation of the ellipse
View attachment 3125
Is this even necessary? Not necessary when using Green's theorem - you would need it to find the line integral. I don't really know how to make use of this when using Green's theorem.
With the theorem I calculated that I'll need to integrate the double integral of 2. Is it really that simple!!

For the limits of the integrals I thought about this:
Making use of the ellipse equation, I said that x= +/- 2*sqrt(1-y²). Those would be the integration limits for dx Looking at that, for dy= +/- 1? Yes

I'd greatly appreciate your help with this!

Thanks

I want to look at the integrand. Hmmm. Looks like you are right!

\(\displaystyle \dfrac{\partial P}{\partial y} = -1\)

\(\displaystyle \dfrac{\partial Q}{\partial x} = 1\)

\(\displaystyle \dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y}= 2\)

 

[Melissa00]

Okay Thank you!!