Let *f* be a function on an interval *a*,*b*].

Divide the interval *a*,*b*]*dx* (see here for explanation of *dx*) and divide the region
under *f* into corresponding vertical strips.

Since these subintervals are so thin, the change of *f* within each of
them can be ignored. Therefore the average of *f* can be obtained by
finding the average of heights of all these strips. How many such strips do
we have? This is easy, each strip is *dx* wide and the total must give
the size of the interval *a*,*b*].*f* should be

This leads us to the following definition of the **average** of a given
function, also called its **mean (value)**:

Definition.

Letfbe a Riemann integrable function on[ We define thea,b].averageoffover this interval by

Note that we get

This is interesting for two reasons. First, by the
Mean value Theorem for
integrals, if *f* is continuous, then it must attain its average at
some point of *a*,*b*].

The above equality is also interesting from the geometric point of view. Is
says the following: If we look at the region under the graph of *f* and
flatten its top at the level of the average of *f*, the resulting
rectangle has the same area as the region under the graph of *f*.