Problem with integration

Questions
,
1

Hi !

I am triyng to calculate the volume of the intersection of three cylinders of same radius r with their axis perpendicular.

I know that the formula is V(r) = 8 ( 2 - sqrt[2]) r^3 ( WolframMathWorld → Steinmetz Solid).

The computation with a numerical value (r = 3) give a correct result (see attachment)

But the same computation with no numerical value for r give a wrong result → 8( 1 - sqrt[2]) r^3

Thanks to any help.

L. Rogliano

2

This seems to be a bug in Integrate. You might want to send email to support@wolfram.com
my system is V 9.01, windows 7, 64 bit

ClearAll[x, y, z, r,f];
expr = Boole[x^2 + y^2 <= r^2 && y^2 + z^2 <= r^2 && z^2 + x^2 <= r^2];
f[value_] :=
 {Integrate[expr, {x, -Infinity, Infinity}, {y, -Infinity, Infinity}, {z, -Infinity, Infinity}] /. r -> value,
  NIntegrate[expr /. r -> value, {x, -Infinity, Infinity}, {y, -Infinity, Infinity}, {z, -Infinity, Infinity}]
  }

now

f[3]
(*  {-8 (-27 + 27 Sqrt[2]), 126.5298722} *)
N[%]
(*  {-89.47012947, 126.5298722}  *)
3

I work with Matematica V 9.0.1 on a iMac late 2013 with OS X 9.0.3 (Mavericks) and 32 Go.
I have sent an email to support@wolfram.com with no significant result.

4

This bug appears to be fixed in the development version of Mathematica 10.