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} *)
