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单词 AreaOfPlaneRegion
释义

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.

Remarks

  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

    0=12P(xdy+ydx).

    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.

    A=abf(x)𝑑x,
    A=12φ1φ2[r(φ)]2𝑑φ,

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

Example.  The ellipse  x2a2+y2b2=1  has the parametric   x=acost,  y=bsint  (0t<2π).  We have

xdy-ydx=[acostbcost+bsintasint]dt=abdt,

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

A=12ab02π𝑑t=πab.
Titlearea of plane region
Canonical nameAreaOfPlaneRegion
Date of creation2013-03-22 15:17:46
Last modified on2013-03-22 15:17:46
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id14
Authorpahio (2872)
Entry typeTopic
Classificationmsc 26B20
Classificationmsc 26A42
Synonymplanar area
Related topicArea2
Related topicDefiniteIntegral
Related topicPolarCurve
Related topicRiemannMultipleIntegral
Related topicPropertiesOfEllipse
Related topicAreaBoundedByArcAndTwoLines
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更新时间:2025/5/4 17:15:19