1. ## Green's theorem

Hey, I have a problem with solving this:

With Green's theorem solve this problem:

2. ## Re: Green's theorem

consider the curve $C$ consisting of

the line segment from (0,0) to $A$, the curved line from $A$ to $B$ and the segment from $B$ to $(0,0)$

and the region $D$ enclosed by $C$ and apply Green's Theorem

3. ## Re: Green's theorem

I tried this, and I don't know how to continue...

4. ## Re: Green's theorem

The path $L_1,L_2,L_3,L_4$ leads to the integral

$$\int _0^{\sqrt{3}}\int _0^{\sqrt{4-y^2}}3\left(x^2+y^2\right)dxdy$$

which is a little difficult to evaluate

on the other hand the path $A\longrightarrow B\longrightarrow O\longrightarrow A$

will result in an integral which is easier to evaluate (using polar coordinates)

$$\int _0^{\pi /3}\int _0^23r^3drd\theta$$

Spoiler:
Final answer: $$4 \pi -\frac{\sqrt{3}}{2}$$