Evaluate the surface integral of the function g over the surface S.
-$G ( x , y , z ) = x ^ { 3 } y ^ { 3 } z ^ { 3 } ; S$ is the surface of the rectangular prism formed from the planes $x = \pm 3 , y = \pm 3$, and $z = \pm 3$
A) $\frac { 531441 } { 4 }$
B) 0
C) $\frac { 19683 } { 16 }$
D) $\frac { 531441 } { 16 }$

Question

Quizplus · Accepted Answer

The function $G(x, y, z) = x^3y^3z^3$ is odd in each of its variables. When integrating over a symmetric domain like the one described (a rectangular prism centered at the origin with sides parallel to the coordinate planes), the contributions from opposite faces cancel out due to the symmetry and the odd nature of the function. Therefore, the integral over the entire surface is 0.

Evaluate the Surface Integral of the Function G Over the Surface