Can we use Green's Theorem to find line integral where circles have different centres?

Question

Suppose we have to find a line integral where $C$ is the boundaries of two circles with different centres. Can we use Green's Theorem here?

Since centres are different its difficult for me to convert to polar coordinates.

Answer 1

Yes, Green's theorem applies here.

Red circle: $x^2+y^2=16$

Blue circle: $x^2+(y-1)^2=4$

By Green's theorem,

$$\oint_C P(x,y)\,\mathrm dx+Q(x,y)\,\mathrm dy=\iint_D\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\,\mathrm dx\,\mathrm dy$$

where $C$ is the union of the red path with counter-clockwise orientation and the blue path with clockwise orientation, and $D$ is the region between the two circles.

In order to compute the integral over $D$, you can first integrate over the interior of the red circle with standard polar coordinates, using

$$\begin{cases}x=r\cos\theta\\y=r\sin\theta\end{cases}$$

then subtract the integral over the interior of the blue circle, using

$$\begin{cases}x=r\cos\theta\\y=r\sin\theta+1\end{cases}$$

For example, if $P(x,y)=-\frac y2$ and $Q(x,y)=\frac x2$, Green's theorem asserts that the line integral over $C$ is equal to the area of the region between the two circles, which is just the area of the larger circle minus the area of the smaller circle, giving a net result of $16\pi-4\pi=12\pi$.