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[parent] area of plane region (Topic)

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

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

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

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
    $\displaystyle 0 = \frac{1}{2}\oint_P (x\,dy+y\,dx).$
    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. The formulae (1) and (2) contain all other formulae concerning the planar area computing, e.g.
    $\displaystyle A = \int_a^b f(x)\,dx,$
    $\displaystyle A = \frac{1}{2}\int_{\varphi_1}^{\varphi_2}[r(\varphi)]^2\,d\varphi,$
    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

$\displaystyle x\,dy-y\,dx = [a\cos{t}\cdot b\cos{t}+b\sin{t}\cdot a\sin{t}]dt = ab\,dt,$
and hence (1) gives for the area of the ellipse
$\displaystyle A = \frac{1}{2}ab\int_0^{2\pi}dt = \pi ab.$



"area of plane region" is owned by pahio.
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See Also: area, definite integral, polar curve, Riemann multiple integral

Other names:  planar area

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Cross-references: ellipse, equation, Green's theorem, positive, line integral, area, closed curve, region, contour
There are 2 references to this entry.

This is version 8 of area of plane region, born on 2005-05-21, modified 2007-04-21.
Object id is 7094, canonical name is AreaOfPlaneRegion.
Accessed 2899 times total.

Classification:
AMS MSC: 26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type)
 26B20 (Real functions :: Functions of several variables :: Integral formulas )
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