Find the Average Value Of $f ( x , y , z ) = x ^ { 3 } + y ^ { 2 } + z ^ { 4 } \text { over the region } Q \text {, where } Q \text { is a }$

Multiple Choice

Find the average value of
$f ( x , y , z ) = x ^ { 3 } + y ^ { 2 } + z ^ { 4 } \text { over the region } Q \text {, where } Q \text { is a }$ cube in the first octant bounded by the coordinate planes, and the planes $x = 4 , y = 1$ , and $z = 2$ . The average value of a continuous function $f ( x , y , z )$ over a solid region $Q$ is $\frac{1}{V} \iiint_{Q} f(x, y, z) d V$ where $V$ is the volume of the solid region $Q$ .

A) $\frac { 293 } { 15 }$
B) $\frac { 31 } { 586 }$
C) $\frac { 293 } { 308 }$
D) $\frac { 308 } { 293 }$
E) $\frac { 15 } { 293 }$