Question

please use completely simplified exact values only, thank you.

8. Evaluate # F. dS if F(x, y, z) = (2x + 24 + xy, z tanh-lz - y - y?, zy+ x +y In sy 2 and S is the boundary of the solid enclosed by the graphs of x = 0, =y?, z=1, and X = 1. # F. dS simply implies that the S is a closed surface, which does not change the question, as you already knew the surface was closed at a glance given your experience in graphing surfaces.

Answer 1

Gauss Divergence Theorem It states that the surface integral of the normal component of a vector function Å taken over closed pulace 's' is equal to the volume volume integral of the divergence of that vector function F taken over volume enclosed by closed surface s. $ kinds the le. SSS (G. P) av $ V Solution: F(x, y, z) = (2 2x + 2 tay, z tanh'z-y-y², zyt xty enxy Z - by s is boundary of solid enclosed x=09 z = y² z=1 and x=1 Since the surface closed Gauss Divergence Theorem given is we apply o By Gauss Divergence Theorem & F. ds SSI (div f) dv - SSS (E.F) du F(x, y, z) = (Fi, F2, F3 ) divF = OF afz ay af2 ƏZ ax = 2 ( 2x + 2y + xy) + 2 (z tonn 'z-y-y2) ta (zyt z xty laxy az

i divf = (2+ y) + (-1-2y) + (4-1) = 2+ y - 1 - 2y + y-1 $ Fids = Isso dv S >> F.ds = 0 s