Draw the annular region bounded by r = 1 and r = 2. Divide it into two regions, each of which is suitable for the application of Green’s Theorem.

We need to draw the annular region bounded by $r=1$and $r=2$and then divide it into two regions, each of which is suitable for the application of Green’s Theorem.

Draw the annular region bounded by $r=1$ and $r=2$ as shown in the following figure:

Here, Q is the annular region described as follows:

$Q=\left\{\right(r,\theta )\mid 1\le x\le 2,0\le \theta \le 2\pi \}.$

Represent this annular region as the union of $R_1$and $R_2$, where each of the sub-regions has a piecewise-smooth simple closed boundary, and so each of which is suitable for the application of Green's Theorem.

Divide this region into two regions as shown in the following figure:

Notice that, the inner curve D is parametrized in the clockwise direction, and the outer edge C is parametrized in the counterclockwise direction.

Then, by Green's Theorem,

${\int }_C \mathbf{F}(x,y)\cdot d\mathbf{r}+{\int }_D \mathbf{F}(x,y)\cdot d\mathbf{r}={\iint }_Q \left(\frac{\mathrm{\partial }{F}_2}{\mathrm{\partial }x}-\frac{\mathrm{\partial }{F}_1}{\mathrm{\partial }y}\right)dA.$