Let the contour of the region in the $xy$ -plane be a closed curve $P$ . Then the area of the region equals to the line integral

(1)

Remarks

1. The formula (1) can be gotten as a special case of Green's theorem by setting $\vec{F} := \frac{1}{2}(-y,\,x)$ .
2. Because $x\,dy+y\,dx = d(xy)$ , we have $$0 = \frac{1}{2}\oint_P (x\,dy+y\,dx).$$ This equation may be added to or subtracted from (1), giving the alternative forms

   (2)

   
   
3. The formulae (1) and (2) contain all other formulae concerning the planar area computing, e.g. $$A = \int_a^b f(x)\,dx,$$ $$A = \frac{1}{2}\int_{\varphi_1}^{\varphi_2}[r(\varphi)]^2\,d\varphi,$$ the former of which is factually same as the latter form of (2).

Example. The ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ has the parametric presentation $x = a\cos{t}$ , $y = b\sin{t}$ ($0 \leqq t < 2\pi$ ). We have $$x\,dy-y\,dx = [a\cos{t}\cdot b\cos{t}+b\sin{t}\cdot a\sin{t}]dt = ab\,dt,$$ and hence (1) gives for the area of the ellipse $$A = \frac{1}{2}ab\int_0^{2\pi}dt = \pi ab.$$