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Integration error in 5.0?


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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