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 > >