Find $\iint_C y dA$ using Greens Theorem where C is a parametric curve

Question

$\iint_C y dA$ where $C$ is the region inside the curve $\vec x (t)=(t-t^3, t^2)$ for $1\le t\le1$

Given $$\int Pdx+Qdy = \iint(\frac{\delta Q}{\delta x}-\frac{\delta P}{\delta y})$$ I could use $P=(x)\quad$ and $\quad Q=(xy)$ to get $\iint_C y dA$

Not sure now on how to plug the parametric curve in the integral

Answer 1

Let's use $P=0$ and $Q = xy$ as you suggest. Then, if we let $C = \partial \Omega$, then $$ \begin{split} \iint_\Omega ydA &= \iint_\Omega \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dA \\ &= \int_{\partial \Omega} P(x,y) dx + Q(x,y)dy \\ &= \int_C xydy \\ &= \int_{t=-1}^{t=1} x(t) y(t) y'(t) dt \end{split} $$ Can you finish this?