area bounded by arc and two lines
Let be the equation of a continuous curve in polar coordinates and be the area of the planar region by the curve and the line segments from the origin to two points of the curve corresponding the polar angles and (). Then the area can be calculated from
Proof. We fit between and a set of values
and denote , and think the line segments from the origin to each point of the curve corresponding the values . Then the region is divided into parts. For every part we form inscribed and circumscribed circular sector with the common tip in the origin and the radii along the lines . 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 means the least and the greatest value of on the interval . Hence the area is between these sums for any division of the interval 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 enclosed by the lemniscate of Bernoulli .
and therefore the whole area in question is .
Example 2. Determine the area enclosed by the logarithmic spiral and two radii and (, ).
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|
|Date of creation||2013-03-22 19:05:15|
|Last modified on||2013-03-22 19:05:15|
|Last modified by||pahio (2872)|
|Synonym||area in polar coordinates|