# 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

 $\displaystyle A\;=\;\frac{1}{2}\oint_{P}(x\,dy-y\,dx)$ (1)

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

Remarks

1. 1.

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

2. 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

 $\displaystyle A\;=\;\oint_{P}x\,dy\;=\;-\oint_{P}y\,dx.$ (2)
3. 3.

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

 $A\;=\;\int_{a}^{b}f(x)\,dx,$
 $A\;=\;\frac{1}{2}\int_{\varphi_{1}}^{\varphi_{2}}[r(\varphi)]^{2}\,d\varphi,$

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   $x=a\cos{t}$,  $y=b\sin{t}$  ($0\leqq t<2\pi$).  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

 $A\;=\;\frac{1}{2}ab\!\int_{0}^{2\pi}\!dt\;=\;\pi ab.$
 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