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Integrals involving square roots

• To: mathgroup at smc.vnet.net
• Subject: [mg73854] Integrals involving square roots
• From: "Andre Giroux" <giroux at dms.umontreal.ca>
• Date: Fri, 2 Mar 2007 06:27:45 -0500 (EST)

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!