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