Chapter 14.5, Problem 36ES

The average value or mean value of a continuous function $f\left(x,y,z\right)$ over a solid G is defined as

$f_{\text{ave}}=\frac{1}{V\left(G\right)}{\displaystyle \underset{G}{\iiint }f\left(x,y,z\right)dV}$

Where $V\left(G\right)$ is the volume of the solid G (compare to the definition preceding Exercise 61 of Section 14.2). Use this definition in these exercises.

Let $d\left(x,y,z\right)$ be the distance from the point $\left(z,z,z\right)$ to the point $\left(x,y,0\right).$ Use the numerical triple integral operation of a CAS to approximate the average value of d for $0\le x\le 1,0\le y\le 1,\text{and}0\le z\le 1.$ Write a short explanation as to why this value may be considered to be the average distance between a point on the diagonal from $\left(0,0,0\right)\text{to}\left(1,1,1\right)$ and a point on the face in the $xy\text{-plane}$ for unit cube $0\le x\le 1,0\le y\le 1,$ and $0\le z\le 1.$