Question

Evaluate the surface integral ∫∫ (3x i + 2y j). dS, where S is the sphere given by x^2 + y^2 + z^2 = 9.

(a) 120π

(b) 180π

(c) 240π

(d) 300π

This question was addressed to me during a job interview.

My enquiry is from Gauss Divergence Theorem topic in division Vector Calculus of Electromagnetic Theory

Answer 1

Correct answer is (b) 180π

To elaborate: We could parameterise surface and find surface integral, but it is wise to use divergence theorem to get faster results. The divergence theorem is given by ∫∫ F.dS = ∫∫∫ Div (F).dV

Div (3x i + 2y j) = 3 + 2 = 5. Now the volume integral will be ∫∫∫ 5.dV, where dV is the volume of the sphere 4πr^3/3 and r = 3units.Thus we get 180π.