Area of a GREEN-region

a) Show that the area of GREEN-region B (which is defined by: $A\subseteq R^2$, $A=B_1\cup ...\cup B_m$, and $\mathring{B}_i\cap \mathring{B}_j=\emptyset$) in the plane is defined by:

$v_2(B)=\oint_{\partial B}xdy=-\oint_{\partial B}ydx=\frac{1}{2}\oint_{\partial B}xdy-ydx$ .

b) With the results from a) compute the content $v_2(B)$ of the domain $B_n\subseteq R^2$ within the curve

$\frac{\sqrt[n]{x^2}}{a^2}+\frac{\sqrt[n]{y^2}}{b^2}=1$ for $n=1,2,3,...$.

My teacher was sick the past week, which is why he will explain Green-Theorem and Divergence-Theorem (Gauss's theorem) on friday, but we have to do exercises which partially deal with Gauss's integration and Green-theorem and hand them in on friday. I tried learning about those in our textbook, but I don't really get it yet.

I found a formula under $Gauss's~ integration$ which is:

$\oint_{\partial A}\rho d\sigma :=\sum_{j=1}^m\oint_{S_j}\rho d\sigma$ .

It says that $\partial A$ is made out of a finite number of smooth parameterized surfaces $S_1,...,S_m$. But I don't know how to get from that to the desired equation in a). (The corrector said that it's enough to show for $n=1,2,3,4,5,6$.

My thoughts on b): I tried looking at it as an ellipse at first, but apparently that's only the case for $n=2$. I suppose I would need to use $Gauss's~integration$ to solve this, but I don't even know what $\rho$ would be in this case.

Anyone got some tips on it? And maybe some short explanation about Gauss's theorem. It's kind of hard to understand just by reading it.