Question

Find the average value of the function \({\rm{f(x, y, z) = x y z}}\) over the cube with side length that lies in the first octant with one vertex at the origin and edges parallel to the coordinate axes.

Short answer

The average value of the function \({\rm{f(x, y, z) = x y z}}\) is \({{\rm{f}}_{{\rm{avg}}}}{\rm{ = }}\frac{{{{\rm{L}}^{\rm{3}}}}}{{\rm{8}}}\).

Step-by-step solution

Concept Introduction

Triple integrals are the three-dimensional equivalents of double integrals. They're a way to add up an unlimited number of minuscule quantities connected with points in a three-dimensional space.

Find average value of the function

Over an area E, the average value of the function f(x, y, z) is given by

\({{\text{f}}_{\text{avg }}}\text{=}\frac{\text{1}}{\text{V}}\iiint_{\text{E}}{\text{f}}\text{(x,y,z)dV}\)

Where V is the volume of the region E, calculated as follows:

\(\text{V=}\iiint_{\text{E}}{\text{d}}\text{V}\)

Find the average value of the function

The region inside the given cube can be defined as

\({\rm{E = \{ (x,y,z)}}\mid {\rm{0}} \le {\rm{x,y,z}} \le {\rm{L\} }}\)

It's also worth noting that the cube's volume is\({\rm{V = }}{{\rm{L}}^{\rm{3}}}\).

Therefore,

\(\begin{aligned}{{\rm{f}}_{{\rm{avg}}}}\rm&= \frac{{\rm{1}}}{{{{\rm{L}}^{\rm{3}}}}}\int_{\rm{0}}^{\rm{L}} {\int_{\rm{0}}^{\rm{L}} {\int_{\rm{0}}^{\rm{L}} {\rm{x}} } } {\rm{yzdxdydz }}\\\rm &= \frac{{\rm{1}}}{{{{\rm{L}}^{\rm{3}}}}}\left( {\int_{\rm{0}}^{\rm{L}} {\rm{x}} {\rm{dx}}} \right)\left( {\int_{\rm{0}}^{\rm{L}} {\rm{y}} {\rm{dy}}} \right)\left( {\int_{\rm{0}}^{\rm{L}} {\rm{z}} {\rm{dz}}} \right)\end{aligned}\)

Keep in mind:

\(\int_{\rm{a}}^{\rm{b}} {\rm{f}} {\rm{(x)dx = }}\int_{\rm{a}}^{\rm{b}} {\rm{f}} {\rm{(z)dz}}\)

Hence,

\(\begin{aligned}{{\rm{f}}_{{\rm{avg }}}}\rm &=\frac{{\rm{1}}}{{{{\rm{L}}^{\rm{3}}}}}{\left( {\int_{\rm{0}}^{\rm{L}} {\rm{x}} {\rm{dx}}} \right)^{\rm{3}}}\\{{\rm{f}}_{{\rm{avg}}}}\rm &= \frac{{\rm{1}}}{{{{\rm{L}}^{\rm{3}}}}}{\left( {\left( {\frac{{{{\rm{x}}^{\rm{2}}}}}{{\rm{2}}}} \right)_{\rm{0}}^{\rm{L}}} \right)^{\rm{3}}}\\{{\rm{f}}_{{\rm{avg}}}}\rm &= \frac{{\rm{1}}}{{{{\rm{L}}^{\rm{3}}}}}{\left( {\frac{{{{\rm{L}}^{\rm{2}}}}}{{\rm{2}}}} \right)^{\rm{3}}}\\{{\rm{f}}_{{\rm{avg}}}}\rm &=\frac{{\rm{1}}}{{{{\rm{L}}^{\rm{3}}}}}{\rm{ *}}\frac{{{{\rm{L}}^{\rm{6}}}}}{{\rm{8}}}{\rm{ = }}\frac{{{{\rm{L}}^{\rm{3}}}}}{{\rm{8}}}\end{aligned}\)

Therefore, the average value of the function \({\rm{f(x, y, z) = x y z}}\) is \({{\rm{f}}_{{\rm{avg}}}}{\rm{ = }}\frac{{{{\rm{L}}^{\rm{3}}}}}{{\rm{8}}}\).