6.4 Average Value of a Function

Mathematica script by Chris Parrish,

cparrish@sewanee.edu

Sources and references for some of these problems include

James Stewart, "Calculus: Concepts and Contexts," Second Edition, Brooks/Cole, 2001

Deborah Hughes-Hallett, Andrew M. Gleason, et. al., "Calculus," Second Edition, John Wiley & Sons, 1998

Robert Fraga, ed., "Calculus Problems for a New Century," The Mathematical Association of America, 1993

Average Value of the Sin over the Interval [0,π]

What do we mean by the "average value of the Sin" over the interval [0,π]?

In[338]:=

In[340]:=

In order to estimate the average value of the Sin over the interval [0,π] ,we divide the interval [0,π] into n equal subintervals, and average the values of the Sin at those points.

In[343]:=

These estimates of the average value of the Sin over the interval [0,π] should improve as n gets larger and larger.

Now calculate the limit of the above expression as n goes to infinity.

In[344]:=

Out[344]=

Hmm. Mathematica needs a little help here.

In[345]:=

Out[345]=

With just a little symbolic manipulation, we can recognize the above limit as an integral.

Since Δx = (b - a)/n with a = 0 and b = π, we replace n with (b - a)/Δx in the expression for approx[n].

In[346]:=

This last expression is a Riemann Sum , and its limit as n goes to infinity is the following integral.

In[348]:=

Out[348]=

All of this leads to our taking the following expression as the definition of the average value of the Sin over the interval [0,π].

In[349]:=

Out[349]=

Is the following illustration at all persuasive?

If the region under the green curve were ice, and it melted, would the new water level be at the red line?

In[350]:=

Created by Mathematica (April 22, 2004)