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MathGroup Archive 2007

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Re: Integrals involving square roots

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74008] Re: Integrals involving square roots
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Mon, 5 Mar 2007 04:56:07 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <es90ol$2lt$1@smc.vnet.net>

In article <es90ol$2lt$1 at smc.vnet.net>,
 "Andre Giroux" <giroux at dms.umontreal.ca> wrote:

> Hi All.
> Mathematica is very unreliable when it computes integrals involving square 
> roots. The following were obtained running Mathematica 5.2 and 5.1 with 
> Windows XP pro:
> 
> In[1]:=
> \!\(\[Integral]\_\(-3\)\%3\(\[Integral]\_\(-\@\(9 - x\^2\)\)\%\(\@\(9 - 
> \x\^2\)\)\(\[Integral]\_0\%\(y - 3\)2 y \[DifferentialD]z \[DifferentialD]y 
> \[DifferentialD]x\)\)\)
> Out[1]=
> \!\(\(81\ \[Pi]\)\/2\)
> In[5]:=
> \!\(\[Integral]\_0\%2\(\[Integral]\_\(-R\)\%R\( Rz\^2\/\@\(R\^2 - x\^2\)\) 
> \\[DifferentialD]x \[DifferentialD]z\)\)
> Out[5]=
> \!\(\(-2\)\ \[Pi]\ Rz\^2\)
> 
> In both cases, the sign is wrong and, with the first integral, it is not 
> obvious. Interestingly, Mathematica 5.0 computes these integrals correctly. 
> So beware!

More than not obvious. I assume that the first integral is

  Integrate[2 y, {x, -3, 3}, {y, -Sqrt[9 - x^2], Sqrt[9 - x^2]}, 
    {z, 0, y - 3}]

for which Mathematica returns.

  (81 Pi)/2

As far as I can see, this integral _is_ correct. It certainly agrees 
with the numerical integral:

  NIntegrate[2 y, {x, -3, 3}, {y, -Sqrt[9 - x^2], Sqrt[9 - x^2]}, 
    {z, 0, y - 3}]

  127.23450236818807

For the second integral, there is no space between R and z^2 in your 
input expression. Moreover, telling Mathematica that R > 0, yields the 
correct answer:

  Assuming[R > 0, Integrate[(R z^2)/Sqrt[R^2 - x^2],    
    {z, 0, 2}, {x, -R, R}]]

  (8 Pi R)/3

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)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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