Use the Divergence Theorem to compute for and Q bounded by and the coordinate planes.
B) 0
D) 4

Question

Use the Divergence Theorem to compute  for  and Q bounded by  and the coordinate planes.
B) 0 
D) 4

Quizplus · Accepted Answer

By the Divergence Theorem, we have

∫∫(F·n)dS = ∫∫∫(∇·F)dV

where F = (x^2yz, x+y, z), n is the outward unit normal vector to the surface Q, and V is the solid bounded by Q and the coordinate planes.

Note that ∇·F = 2xyz + 1, so

∫∫∫(∇·F)dV = ∫∫∫(2xyz+1)dV

Using the limits of integration provided, we have

∫∫∫(2xyz+1)dV = ∫0^1 ∫0^(1-z) ∫0^(1-xy) (2xyz+1)dx dy dz

Evaluating this triple integral gives

∫∫∫(2xyz+1)dV = 0

Therefore, the surface integral ∫∫(F·n)dS is also 0. Thus, the answer is B.

Use the Divergence Theorem to Compute for and Q