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MathGroup Archive 2005

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Re: Integrate is driving me crazy, please help!

  • To: mathgroup at smc.vnet.net
  • Subject: [mg56193] Re: [mg56184] Integrate is driving me crazy, please help!
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Tue, 19 Apr 2005 04:54:49 -0400 (EDT)
  • Reply-to: hanlonr at cox.net
  • Sender: owner-wri-mathgroup at wolfram.com

Use square brackets for all functions

asmp={Element[{a,f,f0,hilla,hillb}, Reals], 
      a>0,hillb>0};

soln = Simplify[Integrate[
      (hillb*((f0+hilla)/
                (2*Pi*f*a*Cos[ArcSin[x/a]]+hillb)))-
        hilla,{x,-a,a}],asmp]

(1/(2*f*Pi*Sqrt[hillb^2 - 4*a^2*f^2*Pi^2]))*
  (2*(f0 + hilla)*ArcTan[(2*a*f*Pi)/Sqrt[hillb^2 - 4*a^2*f^2*Pi^2]]*hillb^2 - 
   I*(f0 + hilla)*Log[-(I/(2*Sqrt[hillb^2 - 4*a^2*f^2*Pi^2]))]*hillb^2 + 
   I*f0*Log[I/(2*Sqrt[hillb^2 - 4*a^2*f^2*Pi^2])]*hillb^2 + 
   I*hilla*Log[I/(2*Sqrt[hillb^2 - 4*a^2*f^2*Pi^2])]*hillb^2 + 
   f0*Pi*Sqrt[hillb^2 - 4*a^2*f^2*Pi^2]*hillb + hilla*Pi*Sqrt[hillb^2 - 
4*a^2*f^2*Pi^2]*
    hillb - 4*a*f*hilla*Pi*Sqrt[hillb^2 - 4*a^2*f^2*Pi^2])

Simplify[soln/.{hilla->3,hillb->50,f0->8,a->1,f->1}]

269 - 6875/Sqrt[625 - Pi^2] + (13750*ArcTan[Pi/Sqrt[625 - Pi^2]])/
(Pi*Sqrt[625 - Pi^2])

%//N

14.0372


Bob Hanlon

> 
> From: Jim Martin <jim.martin at utah.edu>
To: mathgroup at smc.vnet.net
> Date: 2005/04/18 Mon AM 03:08:44 EDT
> To: mathgroup at smc.vnet.net
> Subject: [mg56193] [mg56184]  Integrate is driving me crazy, please help!
> 
> Hello Mathematica Experts:
> 
> I am a biomechanist and work mostly in the area of muscle contraction. I 
> do a lot of numerical computations using excel, but right now I need an 
> analytical solution that represents force as a function of position 
> integrated over a shortening amplitude. I downloaded a trial version of 
> Mathematica and have tried to obtain a solution for this:
> 
> Integrate[(hillb*((f0 + hilla)/(2*pi*f*a*Cos(ArcSin(x/a)) + hillb))) - 
> hilla, {x, -a, a}]
> 
> Mathematica returns this:
> (-4 a ArcSin Cos f hilla pi + (f0 + hilla) hillb (-Log[hillb - 2 a 
>      ArcSin Cos f pi] + Log[hillb + 2 a ArcSin Cos f pi]))/(4 a ArcSin 
> Cos f pi)
> 
> I know the line wrap makes this hard to read so please feel free to 
> email me and I can send you the output as a picture.
> 
> In a sample data set, hilla=3, hillb=50, f0=8, a=1, f=1
> 
> I can numerically integrate this function and obtain a value for that 
> sample data set of 14.04. When I put those sample values into the 
> solution that Mathematica produces, I get 10.01.
> 
> Can any of you please give a hand here? I must be making some simple 
> Mathematica-beginner error but I just can't see it.
> 
> In Mathematica, Log is Log to base e, right (LN in excel)? Did I use 
> variables that have intrinsic functions in Mathematica? Maybe I am 
> misunderstanding the output with regard to implicit parentheses etc. 
> Any help appreciated!
> 
> Thanks,
> 
> Jim
> 
> 
> 


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