Gauss-Green Application on a given integral I'm trying to solve an application of the Gauss-Green theorem, but I can't find a useful example on my textbook...
The problem asks to calculate and prove the Gauss-Green theorem on the double integral
$ \int\int_{}^{} \frac{y}{\sqrt{x^2+y^2}} $
in a domain limited by 
$ 4 ≤ x^2 + y ^2 ≤ 16 $
$ y ≤ x $
$ y ≤ \frac{x}{\sqrt{3}} $
I really don't know where to start... I only know that the formula to apply is 
$ \int\int Qx - Py = \oint  Pdx + Qdy $
and i should consider the given double integeral as the first element of the equation, but how should i proceed to apply and demonstrate Gauss-Green?
 A: The Green-Gauss theorem states
$$\int\!\!\!\int\limits_A {\left( {{{\partial Q} \over {\partial x}} - {{\partial P} \over {\partial y}}} \right)da}  = \int\limits_{\partial A} {Pdx + Qdy} $$
Choose $Q=0$. Then you have
$$\int\!\!\!\int\limits_A { - {{\partial P} \over {\partial y}}da}  = \int\limits_{\partial A} {Pdx} $$
Now in order to relate this to your question, you should find a $P$ such that 
$$ - {{\partial P} \over {\partial y}} = {y \over {\sqrt {{x^2} + {y^2}} }}$$
The following $P$ will do this
$$P =  - \sqrt {{x^2} + {y^2}} \,$$
Now by the green theorem we must have
$$\int\!\!\!\int\limits_A {{y \over {\sqrt {{x^2} + {y^2}} }}da}  = \int\limits_{\partial A} { - \sqrt {{x^2} + {y^2}} dx} $$
Now, I think your question wants to compute each side of the above equation directly and then verifying the green theorem for this special case. So you must compute a line integral and a double integral for this purpose. 
My final hint is to compute these integrals in polar coordinate because your $A$ (domain) has a simple representation in polar coordinates
$$\left\{ \matrix{
  2 \le r \le 4 \hfill \cr 
  {\pi  \over 6} \le \theta  \le {\pi  \over 4} \hfill \cr}  \right.$$
I will leave the computation details for you just saying that each of those integrals are equal to $3\left( {\sqrt 3  - \sqrt 2 } \right)$.
