area under a parametric curve

*To*: mathgroup at smc.vnet.net*Subject*: [mg74983] area under a parametric curve*From*: Roger Bagula <rlbagula at sbcglobal.net>*Date*: Fri, 13 Apr 2007 02:08:33 -0400 (EDT)

In working with unit square "curves" I came up with the idea of using a Lemnicape ( quarter of a curve): x = Sqrt[Abs[Cos[t]]]*Sin[t] y = Sqrt[Abs[Cos[t]]]*Cos[t] x1 = x*Sin[ArcCos[-1/2^((2/3))]] + 1/2 y1 = y + 1/2 Solve[1/2 + Sqrt[Abs[Cos[t]]Cos[t] == 0, t] ParametricPlot[{x1, 2*y1}, {t, 0, 2*Pi}, PlotRange -> {{0, 1}, {0, 1}}] My question is how to get the area under such a curve? Since the curve is symmetrical only {x,0,1/2} is actually necessary. A second question is how to get rid of the curve "in" part that comes about because the Lemnicape isn't exactly symmetrical: what is the maximum x to y instead of the "2" I've used in this as the scaling. It appears to be about 2.1 instead of 2. Here a try at a better curve: x = Sqrt[Abs[Cos[t]]]*Sin[t] y = Sqrt[Abs[Cos[t]]]*Cos[t] x1 = x*Sin[2.226604043204] + 1/2 y1 = y + 1/2 Solve[1/2 + 2.1 Sqrt[ Abs[Cos[t]] Cos[t]/2 == 0, t] ParametricPlot[{x1, 2.1*y + 1}, {t, 0, 2*Pi}, PlotRange -> {{0, 1}, { 0, 1.0}}]

**Follow-Ups**:**Re: area under a parametric curve***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>