Let S be the boundary of the region V defined by oriented with outward-pointing normal.
Let .
The surface integral is equal to which of the following?
E) None of the above

Question

Let S be the boundary of the region V defined by  oriented with outward-pointing normal.
Let  .
The surface integral  is equal to which of the following?
E) None of the above

Quizplus · Accepted Answer

By Divergence Theorem:

∫∫S F · dS = ∭V ∇ · F dV

Here, F = (y - z) i + (z - x) j + (x - y) k and the region V is defined by 
0 ≤ x ≤ 1, 0 ≤ y ≤ 1 - x^2, 0 ≤ z ≤ 1 - x

∇ · F = ∂(y - z)/∂x + ∂(z - x)/∂y + ∂(x - y)/∂z = -3

∭V ∇ · F dV = -3V where V = ∫∫R (1-x^2) dx dy = 2/3

Therefore, ∫∫S F · dS = -3V = -2

Let S Be the Boundary of the Region V Defined