Learning Objectives
- Calculate the average value of a function of three variables.
Recall that we found the average value of a function of two variables by evaluating the double integral over a region on the plane and then dividing by the area of the region. Similarly, we can find the average value of a function in three variables by evaluating the triple integral over a solid region and then dividing by the volume of the solid.
theorem: average value of a function of three variables
If [latex]f(x, y, z)[/latex] is integrable over a solid bounded region [latex]E[/latex] with positive volume [latex]V(E)[/latex], then the average value of the function is
[latex]{f_{\text{ave}}} = {\frac{1}{V(E)}}\underset{E}{\displaystyle\iiint}{f}{(x,y,z)}{dV}[/latex].
Note that the volume is [latex]{V(E)} = \underset{E}{\displaystyle\iiint}{1}{dV}[/latex].
Example: finding an average temperature
The temperature at a point [latex](x, y, z)[/latex] of a solid [latex]E[/latex] bounded by the coordinate planes and the plane [latex]x+y+z=1[/latex] is [latex]{T}{(x,y,z)} = {(xy+8z+20)}{^\circ}{\text{C}}[/latex]. Find the average temperature over the solid.
try it
Find the average value of the function [latex]f(x, y, z)=xyz[/latex] over the cube with sides of length 4 units in the first octant with one vertex at the origin and edges parallel to the coordinate axes.
Watch the following video to see the worked solution to the above Try It
