# Application of Green's Theorem

• May 20th 2011, 08:01 PM
jonmondalson
Application of Green's Theorem
Hi, I'm trying to answer the following question:

http://imageshack.us/m/849/9551/greenstheorem.png

Basically, I know that you have to use Green's Theorem:

$\displaystyle \displaystyle\oint_C P(x)dx +Q(x)dy = \displaystyle\iint_A (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\ dx\,dy$

Which makes our integral:

$\displaystyle \displaystyle\iint_A x \ dx\,dy$

My problem is, I don't know how to calculate the limits for the area A. Do we use a parameterisation, or are the limits simply the points of intersection of the functions.

Any help is immensely appreciated.

Thanks :)
• May 20th 2011, 08:35 PM
FernandoRevilla
The intersection points of $\displaystyle y=x$ and $\displaystyle y=x^2-2x$ are $\displaystyle (0,0)$ and $\displaystyle (3,3)$ . You'll obtain:

$\displaystyle \displaystyle\oint_C 3xy\;dx +2x^2\;dy=\iint_Ax\;dxdy= \int_0^3\;dx\int_{x^2-2x}^xx\;dy=\ldots$
• May 21st 2011, 02:17 AM
HallsofIvy
IF you were to integrate around the boundary, you would need to parameterize the curve. You don't use parametric equations for an area in 2 dimensions. That's why it is easier to use Green's theorem.