Re: Area of an Epicycloid ???

*To*: mathgroup at smc.vnet.net*Subject*: [mg21397] Re: Area of an Epicycloid ???*From*: "Allan Hayes" <hay at haystack.demon.co.uk>*Date*: Tue, 4 Jan 2000 02:12:30 -0500 (EST)*References*: <84plms$lul@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Steve, Better to use polar coordinates to find the area of the epicycloid x[t_] = (R + r)*Cos[t] - r*Cos[(R + r)*t/r]; y[t_] = (R + r)*Sin[t] - r*Sin[(R + r)*t/r]; (* it usually helps respondents if you give expressions in Mathematica form: for example Cos[t] not cos(t}*) rad[t_] = Sqrt[ x[t]^2 + y[t]^2] // Simplify Sqrt[2*r^2 + 2*r*R + R^2 - 2*r*(r + R)*Cos[(R*t)/r]] area[R_, r_] = Integrate[rad[t]^2/2, {t, 0, 2Pi}] // Simplify (Pi*R*(2*r^2 + 2*r*R + R^2) - r^2*(r + R)* Sin[(2*Pi*R)/r])/R area[1, 1] 5 Pi Check: for very small r we would expect to get very near to the area of the circle: area[1, 1/1000000] (500001000001*Pi)/500000000000 Allan --------------------- Allan Hayes Mathematica Training and Consulting Leicester UK www.haystack.demon.co.uk hay at haystack.demon.co.uk Voice: +44 (0)116 271 4198 Fax: +44 (0)870 164 0565 "Guybrush Threepwood" <guybrush_ at gmx.de> wrote in message news:84plms$lul at smc.vnet.net... > hello all ! > > im just trying to calculate the area between the epicyloid curve and the big > circle ! > the problem is that there are some x coordinates which have 2 y values.... > so i cant just simply integrate ! maybe someone of you has a hint for me ! > Someone told me that maybe polar coordinates could be helpfull... what do ya > think ? > > parameter form formula for epicycloids: > > x(t) = (R+r) * cos (t) - r * cos ((R+r)*t / r ) > y(t) = (R+r) * sin (t) - r * sin ((R+r) *t /r ) > > thanks in advance & bye > > Steve martins ! > > > > > > > > > > >