Integral of $\int y^2 dx + x dy$ using Greens Formula

Question

We have the following integral:

$\int y^2 dx + x dy$

I would like to rewrite this integral using Greens Formula. My attempt at this gives: $\int\int (P'_x - Q'_y)dxdy$ where $(P,Q) = (y^2,x)$, thus resulting in $\int\int 0dxdy$.

This is not correct, how would I do it correctly?

Answer 1

Using Stokes Theorem: $$ \int_{\partial\Sigma} y^2 dx + x dy = \iint_{\Sigma} d\left[y^2 dx + x dy\right]= \iint_{\Sigma} \left[-2y dx\, dy + dx\, dy\right]=\iint_{\Sigma} \left[-2y + 1\right] dx\, dy $$