If S=xyz1≤x2+y2+z2≤4 then ∭SdV=283π

This statement is false.

To see why, we can use spherical coordinates to evaluate the integral:

∭S dV = ∫φ=0π/2 ∫θ=02π ∫r=1√(4-r^2) r^2 sinφ dr dθ dφ

Evaluating this integral gives:

∭S dV = 4π/3 [(4/3)^(3/2) - 1]

This is approximately equal to 16.7553, which is not equal to 283π. Therefore, the statement is false.