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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: [mg56216] Re: Integrate is driving me crazy, please help!
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Tue, 19 Apr 2005 04:55:35 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <d3vn1h$jm9$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <d3vn1h$jm9$1 at smc.vnet.net>,
 Jim Martin <jim.martin at utah.edu> wrote:

> 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}]

Instead of 2*pi*f*a*Cos(ArcSin(x/a)) you mean 

  2 Pi f a Cos[ArcSin[x/a]] + hillb

(In StandardForm, pi needs to be capitalised and square brackets are 
required).

> 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)

In this output you will see expressions like 

  ArcSin Cos f pi

which make no sense. Correcting the integrand, and massaging the result 
from Integrate, I get

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

> In a sample data set, hilla=3, hillb=50, f0=8, a=1, f=1

For this dataset, numerical evaluation gives 14.0372.
 
> I can numerically integrate this function and obtain a value for that 
> sample data set of 14.04. 

But how did you do this and get this correct result if you used the 
wrong integrand?

> 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. 

Wrong type of parentheses. If you use TraditionalForm for input and 
output (Cell | Default Input FormatType, etc.) you _can_ enter 
expressions using () instead of [].

Cheers,
Paul

-- 
Paul Abbott                                   Phone: +61 8 6488 2734
School of Physics, M013                         Fax: +61 8 6488 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul


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