Average value of a function of three variables

 Goal: Find the average value of a function of three variables

The average value of a function of two variables is found by dividing the value of the double integral of the function over the region of the plane by the volume of that region. Similarly, the average value of a function of three variables is found by dividing the value of the triple integral of the function over the solid region by the volume of that region.

Theorem

If a function f(x,y,z) is integrable over a solid bounded region E with positive value V(E), then the average value of f is given by:

Example

Solution

Let's draw the region E:

The plane x + y + z =1 has the following intercepts with the axes: (1,0,0), (0,1,0) and (0,0,1). the region E is then determined by:

Practice