Integration error in 5.0?

*To*: mathgroup at smc.vnet.net*Subject*: [mg48728] Integration error in 5.0?*From*: "Richard Ollerton" <r.ollerton at uws.edu.au>*Date*: Fri, 11 Jun 2004 03:53:37 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

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