Hi,
Have you tried using Ball instead of Sphere?
r0 = Ball[{0, 0, 0}, 1.0]
r1 = ImplicitRegion[x^2 + y^2 + z^2 <= 1., {x, y, z}]
NIntegrate[(x + y + z)^2, {x, y, z} \[Element] r0]
NIntegrate[(x + y + z)^2, {x, y, z} \[Element] r1]
gives the same result (2.51327).
Cheers,
Marco
