area bounded by arc and two lines

Let  r=r(φ)  be the equation of a continuousMathworldPlanetmath curve in polar coordinatesMathworldPlanetmath and A be the area of the planar region by the curve and the line segmentsMathworldPlanetmath from the origin to two points of the curve corresponding the polar anglesMathworldPlanetmath α and β (>α).  Then the area can be calculated from

A=12αβ[r(φ)]2𝑑φ. (1)

Proof.  We fit between α and β a set of values

φ1<φ2<<φn-1 (2)

and denote  α=φ0,  β=φn  and think the line segments from the origin to each point of the curve corresponding the values φi.  Then the region is divided into n parts.  For every part we form inscribedMathworldPlanetmath and circumscribedMathworldPlanetmath circular sector with the common tip in the origin and the radii along the lines  φ=φi.  The union of the inscribed sectors is contained in the region and the union of the circumscribed sectors contains the region.  The unions have the areas


where ri means the least and Ri the greatest value of r(φ) on the intervalMathworldPlanetmathPlanetmath[φi-1,φi].  Hence the area A is between these sums for any division of the interval  [α,β]  with the values of (2).  But by the definition of the Riemann integral we know that there is only one real number having this property for any division and that also the definite integral


is between those sums.  Q.E.D.

Example 1.  Determine the area A enclosed by the lemniscate of Bernoullir=cos2φ.

The portion of the lemniscateMathworldPlanetmath situated in the first quadrantMathworldPlanetmath is gotten when φ gets the values from 0 to π4, whence we have


and therefore the whole area in question is a2.

Example 2.  Determine the area A enclosed by the logarithmic spiralMathworldPlanetmathr=Cekφ  and two radii  r1:=Cekφ1  and  r2:=Cekφ2  (k>0,  φ1<φ2).

The (1) directly yields



  • 1 Ernst Lindelöf: Johdatus korkeampaan analyysiin. Fourth edition. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
  • 2 N. Piskunov: Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele.  Kirjastus Valgus, Tallinn (1966).
Title area bounded by arc and two lines
Canonical name AreaBoundedByArcAndTwoLines
Date of creation 2013-03-22 19:05:15
Last modified on 2013-03-22 19:05:15
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 15
Author pahio (2872)
Entry type Derivation
Classification msc 51-01
Classification msc 53A04
Synonym area in polar coordinates
Related topic SectorOfACircle
Related topic AreaOfPlaneRegion
Related topic SubstitutionNotation