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MathGroup Archive 1999

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Re: Re: Solving difficult integral a follow up

  • To: mathgroup at smc.vnet.net
  • Subject: [mg18747] Re: [mg18665] Re: [mg18620] Solving difficult integral a follow up
  • From: Richard Gass <gass at physics.uc.edu>
  • Date: Sat, 17 Jul 1999 02:36:56 -0400
  • Sender: owner-wri-mathgroup at wolfram.com

>> Date: Thu, 15 Jul 1999 01:45:51 -0400
>> From: Richard Gass <gass at physics.uc.edu>
To: mathgroup at smc.vnet.net
>> To: mathgroup at smc.vnet.net
>> Subject: [mg18747] [mg18665] Re: [mg18620] Solving difficult integral a follow up
>>
>> Christian wrote
>> <I've got the following problem:
>>
>> <I need to solve the following integral:
>>
>> <Integrate[Sqrt[C0+C1*x+C2*x^2+C3*x^3+C4*x^4],{x,0,t}]
>>
>> <For those of you who do not recognize it at once: it's the integral of the
>> <square root of a 4th degree polynomial in x. I've tried to solve the above
>> <integral with Mathematica, but after, say, two hours of computing I stopped
>> <the calculation - it looked like the Kernel crashed or whatever. Is
>>there an
>> <analytical solution to this problem at all?
>>
>> I replied that I could do the indefinite integral but that I thought the
>> definite integral was hopeless because of branch cut problems. However
>> my machine ( a fairly old Power Mac) run 4.0  returns an answer for
>> Integrate[Sqrt[C0 + C1*x + C2*x^2 + C3*x^3 + C4*x^4], {x, 0, t}]. THe
>> result takes a  few minutes or so. I am not at all sure the result is
>> correct.
>> Richard Gass
>> Department of Physics
>> University of Cincinnati
>> Cincinnati, OH 45221
>> phone- 513-556-0519
>> E-Mail gass at physics.uc.edu
>>
>
>My Mac (a G3 that's about a year old) will do the symbolic integral, too.  It
>results in about 13 pages of Root Objects intermixed with Elliptic
>functions and
>Arcsins.  However, when I substitute the value 1 for each of the coefficients
>and try to plot the resulting integral over the interval from 0 to 5,
>Mathematica v4 responds with "not a machine-size real number" messages and
>produces no plot.  This in spite of the fact that 1 + x + x^2 + x^3 + x^4
>is a
>positive real number everywhere.
>
>--Lou Talman
>  Department of Mathematical and Computer Sciences
>  Campus Box 38
>  Metropolitan State College of Denver
>  PO Box 173362
>  Denver CO 80217-3362
>
>  http://clem.mscd.edu/~talmanl

I get the same result that you do when I try this. If I look at the
expression I find that it is full of terms like (-1)^(1/5), (-1)^(2/5) and
so on. I think that they arise because  the "Wrong" root has been picked by
the integration routines . If for example we try Solve[x^5==1] we get five
solutions only one of which is real. I think the integration routines have
picked up one of the complex roots instead of the real root.
Richard Gass
Department of Physics
University of Cincinnati
Cincinnati, OH 45221
phone- 513-556-0519
E-Mail gass at physics.uc.edu


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