If a function *f* is integrable over an interval [*a*,*b*], then we
define the average value of *f*, which we'll denote as ,
on this interval to be

Note that the average value is just a number. Note furthermore that we can rearrange the definition to give

If on [*a*,*b*], then the average value has the following
geometrical interpretation: is the height of a
rectangle of width *b*-*a* such that the area of this rectangle is equal
to the area under the graph of *f* from *a* to *b*. The following
example shows you how to compute an average. The last plot command
shows the function and the top of this rectangle.

> f :=x -> x*sin(x) ;

> plot(f(x),x=0..Pi);

> f_ave := int(f(x),x=0..Pi)/Pi;

> plot({f(x),f_ave},x=0..Pi);

Fri Jan 24 15:07:34 EST 1997