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MathGroup Archive 2004

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Re: Integration error in 5.0?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48842] Re: Integration error in 5.0?
  • From: Urijah Kaplan <uak at sas.upenn.edu>
  • Date: Sat, 19 Jun 2004 04:30:41 -0400 (EDT)
  • Organization: University of Pennsylvania
  • References: <cabql1$ouc$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In[1]:=
4*Integrate[1, {x, 0, 2/Sqrt[5]},
    {y, 0, Sqrt[4/5 - x^2]}, {z, 2*Sqrt[x^2 + y^2],
     Sqrt[4 - x^2 - y^2]}]

Out[1]=
(16/15)*(5 - 2*Sqrt[5])*Pi

on version 5.01


		--Urijah Kaplan


Richard Ollerton wrote:

> I used Mathematica 5.0 (Windows 2000) to find the value of an integral
> representing the volume under the sphere x^2 + y^2 + z^2 = 4 and above the
> cone z = 2 Sqrt(x^2 + y^2) (just to check my own calculation).  The value
> given for the analytical calculation in Cartesian coordinates using
> Integrate is incorrect while exactly the same integral using NIntegrate is
> correct as confirmed by the equivalent version in spherical coordinates.
> 
> 4Integrate[1,{x,0,2/Sqrt[5]},{y,0,Sqrt[4/5-x^2]},{z,2Sqrt[x^2+y^2],Sqrt[4-x^
> 2-y^2]}]
> 
> gives -32Pi/(3Sqrt[5])
> 
> while
> 
> 4NIntegrate[1,{x,0,2/Sqrt[5]},{y,0,Sqrt[4/5-x^2]},{z,2Sqrt[x^2+y^2],Sqrt[4-x
> ^2-y^2]}]
> 
> gives 1.76889
> 
> The equivalent in spherical is correct:
> 
> Integrate[r^2 Sin[p],{t,0,2Pi},{p,0,ArcTan[1/2]},{r,0,2}]
> 
> gives 16(5-2Sqrt[5])Pi/15
> 
> with
> 
> N[%]
> 
> 1.76889
> 
> Has this been noticed previously?
> 
> 
> Richard Ollerton
> 
> 


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