Surface Integrals of Scalar Fields

Question

Evaluate ∫∫_S x²yzdσ, where S is the part of the plane 2x+3y−z = −1 that lies above the rectangle [0, 3] × [0, 2]

Answer 1

∫∫_S x²yzdσ = ∫∫_Tx²y(2x + 3y + 1)√1 + 2² + 3²)dxdy = √14∫₀³∫₀²(2x³y + 3x²y² + x²y)dydx =

√14∫₀³(x³y² + x²y³ + x²y²/2)₀²dx = √14 ∫₀³(4x³ + 10x²)dx = √14 (x⁴ + 10x³/3)₀³ = (81 + 90)√14 = 171√14

T is projection of σ on the z = 0 plane