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]?

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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.

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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.

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Hmm. Mathematica needs a little help here.

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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].

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

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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].

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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?

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Created by Mathematica  (April 22, 2004)