Finding the average value of a function with 2 variables

 We have seen previously that double integrals can be used to calculate volumes and areas. We can also use double integrals to find the average value of a function with 2 variables. 

Definition

If a function f(x,y) is integrable over a plane bounded region D with positive area A(D), then the average value of f is given by:

Note that the area is:

Example

Find the average value of the function f(x,y) = 7xy² on the region bounded by the line y = x and the curve x = ⎷y.

Solution

The figure above represents the region bounded by the line y = x and the curve x =⎷ y

First find the area A(D) where the region D is given by the figure above. We have:

Then the average value of this function over the given region is:

  Practice