Gauss/Divergence Theorem

Question

Calculate ∫∫_S F • dS

F = ⟨ 4x³z, 4y³z , 3z⁴ ⟩

S is the surface of the solid bounded by the hemispheres

z = sqrt(25 - x^2 - y^2), z = sqrt(4 - x^2 - y^2) and the plane z = 0

Answer 1

divF = 12x²z + 12y²z + 12z³;

I = ∫∫_SF•ds = ∫∫∫_v12z(x² + y² + z²)dV; In spherical system: x = rsinθcosφ; y = rsinθsinφ; z = rcosθ;

2 ≤ r ≤ 5; 0 ≤ θ ≤ π/2; 0 ≤ φ ≤ 2π; dV = r²sinθdrdθdφ; ; x² + y² + z² = r²;

I = ∫₂⁵12r⁵dr∫₀^π/2cosθsinθdθ∫₀^2πdφ = r⁶₂⁵(- cos2θ/2)₀^π/2φ₀^2π = (5⁶ - 2⁶)·1·2π = 31122π