A bug-looking behavior during integration

*To*: mathgroup at smc.vnet.net*Subject*: [mg129946] A bug-looking behavior during integration*From*: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>*Date*: Wed, 27 Feb 2013 23:42:14 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net

Dear community, I would like to report a strange behaviour, that looks most of all like a bug. The integral of the UnitStep of some function Integrate[ UnitStep[Sqrt[Sqrt[x^2 + y^2] + x] - 0.7 Sqrt[x^2 + y^2]], {x, -Infinity, Infinity}, {y, -Infinity,Infinity}] 0 returns zero as you see, though the function under the integral is 1 within some domain in the form of a cardioid, and zero outside of it. To make it sure evaluate this: Plot3D[UnitStep[ Sqrt[Sqrt[x^2 + y^2] + x] - 0.7 Sqrt[x^2 + y^2]], {x, -3, 5}, {y, -3, 5}] The integral must be positive, therefore. Taking the numerical value of the same integral one finds a finite positive value: NIntegrate[ UnitStep[Sqrt[Sqrt[x^2 + y^2] + x] - 0.7 Sqrt[x^2 + y^2]], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}] 19.6268 Is it indeed a bug? Regards, Alexei Alexei BOULBITCH, Dr., habil. IEE S.A. ZAE Weiergewan, 11, rue Edmond Reuter, L-5326 Contern, LUXEMBOURG Office phone : +352-2454-2566 Office fax: +352-2454-3566 mobile phone: +49 151 52 40 66 44 e-mail: alexei.boulbitch at iee.lu

**Follow-Ups**:**Re: A bug-looking behavior during integration***From:*Alex Krasnov <akrasnov@eecs.berkeley.edu>