       Re: problem getting the area under a parmetric curve

• To: mathgroup at smc.vnet.net
• Subject: [mg52775] Re: problem getting the area under a parmetric curve
• From: "Roger L. Bagula" <rlbtftn at netscape.net>
• Date: Mon, 13 Dec 2004 04:22:19 -0500 (EST)
• References: <cpauof\$ipf\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

This distribution isn't unique either: I get the sam type of result with
a power five, but the plot isn't complete in this case.
I should mention that an attempt to solve for pair functions of this
type failed  in Mathematica ( gave the wrong functional results).

(* power five  pair and rotated distribution function*)
Clear[x0,y0,ang,x,y,z]
x0=(1-t^5)/(1+t^5)
y0=2^(1/5)*(5 )^(1/5)*t*(1+2 t^10+t^20/5)^(1/5)/(1+t^5)
ParametricPlot[{x0,y0},{t,-2*Pi,2*Pi}]
Simplify[x0^3+y0^3]
ang=4
x=Cos[Pi/ang]*x0-Sin[Pi/ang]*y0
y=Cos[Pi/ang]*y0+Sin[Pi/ang]*x0
NSolve[x==0,t]
N[y/.t->0.586173448422246679]
N[x/.t->0.586173448422246679]
ParametricPlot[{x,y},{t,-2*Pi,2*Pi}]
f[t_]=x
norm=Integrate[-y*f'[t],{t,0.586173448422246679,Infinity}]
a0=N[2*norm]
g1=ParametricPlot[{x,y}/(a0),{t,-2*Pi,2*Pi}]
g2=Plot[Exp[-t^2/2]/Sqrt[2*Pi],{t,-Pi,Pi}]
Show[{g1,g2}]

Roger Bagula wrote:
> I worked on this late last night.
> I had trouble even having the curve well defined
> finding the area under it.
> I used the symmetry to integrate the side
> that was easiest. There is only one real zero for the x parametric
> in t  which helps.
> An Infinite integral does appear to exist for the distribution.
>
> (* cubic pair and rotated distribution function*)
> Clear[x0,y0,ang,x,y,z]
> x0=(1-t^3)/(1+t^3)
> y0=2^(1/3)*t*(3+t^6)^(1/3)/(1+t^3)
> ParametricPlot[{x0,y0},{t,-2*Pi,2*Pi}]
> Simplify[x0^3+y0^3]
> ang=4
> x=Cos[Pi/ang]*x0-Sin[Pi/ang]*y0
> y=Cos[Pi/ang]*y0+Sin[Pi/ang]*x0
> NSolve[x==0,t]
> N[y/.t->0.486313]
> N[x/.t->0.486313]
> ParametricPlot[{x,y},{t,-2*Pi,2*Pi}]
> f[t_]=x
> norm=Integrate[-y*f'[t],{t,0.486313,Infinity}]
> a0=N[2*norm]
> g1=ParametricPlot[{x,y}/(a0),{t,-2*Pi,2*Pi}]
> g2=Plot[Exp[-t^2/2]/Sqrt[2*Pi],{t,-Pi,Pi}]
> Show[{g1,g2}]
> Respectfully, Roger L. Bagula
>
> tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
> alternative email: rlbtftn at netscape.net
> URL :  http://home.earthlink.net/~tftn
>

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