area of plane region

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

$A={\displaystyle \frac{1}{2}}{\displaystyle {\oint }_P}(xdy-ydx)$ (1)

taken in the positive (i.e. anticlockwise) circling direction.

Remarks

1. The (1) can be gotten as a special case of Green’s theorem by setting  $\overrightarrow{F}:=\frac{1}{2}(-y,x)$.

2. Because  $xdy+ydx=d(xy)$,  we have

   $$0=\frac{1}{2}{\oint }_P(xdy+ydx).$$

   This equation may be added to or subtracted from (1), giving the alternative forms

   $A={\displaystyle {\oint }_P}x𝑑y=-{\displaystyle {\oint }_P}y𝑑x.$ (2)

3. The formulae (1) and (2) all other formulae concerning the planar area computing, e.g.

   $$A={\int }_a^{b}f(x)𝑑x,$$

   $$A=\frac{1}{2}{\int }_{{\phi }_1}^{{\phi }_2}{[r(\phi )]}^2𝑑\phi ,$$

   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   $x=a\cos t$,  $y=b\sin t$  ().  We have

$$xdy-ydx=[a\cos t\cdot b\cos t+b\sin t\cdot a\sin t]dt=abdt,$$

and hence (1) gives for the area of the ellipse

$$A=\frac{1}{2}ab{\int }_0^{2\pi }𝑑t=\pi ab.$$

Title	area of plane region
Canonical name	AreaOfPlaneRegion
Date of creation	2013-03-22 15:17:46
Last modified on	2013-03-22 15:17:46
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	14
Author	pahio (2872)
Entry type	Topic
Classification	msc 26B20
Classification	msc 26A42
Synonym	planar area
Related topic	Area2
Related topic	DefiniteIntegral
Related topic	PolarCurve
Related topic	RiemannMultipleIntegral
Related topic	PropertiesOfEllipse
Related topic	AreaBoundedByArcAndTwoLines