I have tried to apply greens theorem with P(x,y)=-y and Q(x,y)=x, and gotten ∫ F • ds = 2*Area(D), where F(x,y)=(P,Q) ===> Area(D) = 1/2 ∫ F • ds = 1/2 ∫ (-y,x) • n ds . This is pretty much the most common approach to an area of region problem. But here they ask you to prove this bizarre relation of Area(D) = 1/2 ∫ F • ds = ∫ (x,y) • n ds. I am clueless what to do. Without a good understanding of part (a) of the question, I don't know how to approach (b) at all. I know the parameterisation could be x=acost , y=bsint, 0≤b≤2π. It seems easy but I am in desperate need of some guidance. Thanks heaps for helping!!