I wish to find out the area enclosed by the ellipse $C:=2x^2+3y^2=2y$ using Green's theorem.
I know how to parametrize the ellipse and understand Green's theorem I just don't understand how it is useful in this case.
Looking at my notes it says $$Area=\int_C x~dy$$ but it isn't at all obvious where this comes from and that this is even true.
Could anyone clarify.
Let $D$ be the region enclosed by the ellipse $C$. Then the area of $D$ is $$\text{Area}(D)=\iint_D \text{ d}A$$
Now we'd like to use Green's theorem to convert this to a line integral along the boundary. Green's theorem states $$\iint_D \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} \text{ d}A=\int_C P\text{ d}x+Q \text{ d}y$$
So we need to find a vector field $F(x,y)=P(x,y)\hat{i}+Q(x,y)\hat{j}$ such that $$\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=1$$ One such vector field is given by $F(x,y)=x\hat{j}$. Hence by Green's theorem $$\iint_D \text{ d}A=\int_C x\text{ d}y$$
