area of plane region

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

A=12P(xdy-ydx) (1)

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


  1. 1.

    The (1) can be gotten as a special case of Green’s theorem by setting  F:=12(-y,x).

  2. 2.

    Because  xdy+ydx=d(xy),  we have


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

    A=Px𝑑y=-Py𝑑x. (2)
  3. 3.

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


    the former of which is factually same as the latter form of (2).

Example.  The ellipsex2a2+y2b2=1  has the parametric   x=acost,  y=bsint  (0t<2π).  We have


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

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