Verified answer

Applying cylindrical coordinates yields

∫∫∫ (y - x)z dV

= ∫(θ = 0 to 2π) ∫(r = 0 to 4) ∫(z = 0 to r sin θ) (r sin θ - r cos θ) z * (r dz dr dθ)

= ∫(θ = 0 to 2π) ∫(r = 0 to 4) ∫(z = 0 to r sin θ) zr^2 (sin θ - cos θ) dz dr dθ

= ∫(θ = 0 to 2π) ∫(r = 0 to 4) (1/2) z^2 r^2 (sin θ - cos θ) {for z = 0 to r sin θ} dr dθ

= ∫(θ = 0 to 2π) ∫(r = 0 to 4) (1/2) (r^2 sin^2(θ)) * r^2 (sin θ - cos θ) dr dθ

= ∫(θ = 0 to 2π) (sin^3(θ) - sin^2(θ) cos θ) dθ * ∫(r = 0 to 4) (1/2) r^4 dr

= ∫(θ = 0 to 2π) [(1 - cos^2(θ)) sin θ - sin^2(θ) cos θ] dθ * ∫(r = 0 to 4) (1/2) r^4 dr

= 0, due to the first factor.

I hope this helps!