So I need to integrate (Using Greens Theorem) $\int_C ydx + xy dy$ over $x=2y^2$ and the straight line along the given path $C$ from $(0,0)$ to $(2,1)$
(a) $x=2y^2$
(b) the straight line path which is $x=2y$, right?
So using Greens Theorem $P(x,y)=y$ and $Q(x,y)=xy$ thus $Q_x-P_y=y-1$
Thus
$$\int_C ydx + xy dy= \int \int y-1 dx dy$$
But I thought for (a) the integration was dx dy where the $x$ bounds are $0$ to $2y^2$ and $y$ bounds from $0$ to $1$ but this doesn't give me the desired answer. Where am I going wrong in choosing the bounds and order of integration?
BTW this is not for homework, Im TAing for Calc III and took it back in 2009 so I forgot a lot of stuff so I'm self teaching myself the material over again as the class goes. I don't need the answer, just a comment on what the bounds and order of integration would be and why.