3336 The average value or mean value of a continuous functionf(x, y, z) over a solid G

Chapter 14, Problem 36

(choose chapter or problem)

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

                                        \(f_{\text {ave }}=\frac{1}{V(G)} \iiint_{G} f(x, y, z) d V\)

where \(V(G)\) 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 \(d(x,y,z)\) be the distance from the point \((z, z, z)\) to the point \((x, y, 0)\). Use the numerical triple integral operation of a CAS to approximate the average value of \(d\) for \(0 \leq x \leq 1,0 \leq y \leq 1\), and \(0 \leq Z \leq 1\). Write a short explanation as to why this value may be considered to be the average distance between a point on the diagonal from \((0, 0, 0)\) to \((1, 1, 1)\) and a point on the face in the \(xy\)-plane for the unit cube \(0 \leq x \leq 1,0 \leq y \leq 1\), and \(0 \leq Z \leq 1\).

Equation Transcription:

(, , )

(, y, 0)

0 ≤  ≤ 1, 0 ≤  ≤ 1

0 ≤  ≤ 1

(0, 0, 0)

(1, 1, 1)

Text Transcription:

f(x,y,z)

G

f_ave=1/V (G) Integral integral integral_Gf(x,y,z)dV

V(G)

d(x,y,z)

(z, z, z)

(x, y, 0)

d

0 <= x <= 1, 0 <= y <= 1

0 <= z <= 1

(0, 0, 0)

(1, 1, 1)

xy