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Definite integrals and average values

If a function f is integrable over an interval [a,b], then we define the average value of f, which we'll denote as tex2html_wrap_inline200 , 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 tex2html_wrap_inline202 on [a,b], then the average value has the following geometrical interpretation: tex2html_wrap_inline200 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);

William W. Farr
Fri Jan 24 15:07:34 EST 1997