Formula for the area of the interior of a closed contour in the complex plane

Question

Let $\gamma \colon [a,b] \to \mathbb{C}$ be closed contour in the complex plane. What is the formula for the area of the interior $I(\gamma)$ of $\gamma$? My assumption is that it involves some kind of contour integral. If possible, please kindly show me a derivation and direct me to a resource where I can research this further. Many thanks.

Answer 1

This does not differ from the case of a parametric curve $x=f(t),y=g(t)$, such that the area can be computed as

$$A=\oint x\,dy=-\oint y\,dx=\frac12\oint(x\,dy-y\,dx).$$

Noticing that $x\,dx+y\,dy$ is a total differential, which integrates to $0$, this explains the formula

$$\frac12\oint \bar z\,dz=iA.$$

Answer 2

Note that applying Green formula for $\bar z dz$ we get that the area inside $\gamma$ (assuming the curve is simple and piecewise smooth say) is $\frac{1}{2i}\int_{\gamma}{\bar z }dz$