Average Function Value
The first application of integrals that we�ll take a look
at is the average value of a function. The following fact tells us how to
compute this.
Average Function Value
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The average value of a function
 over
the interval [a,b] is given by,
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To see a justification of this formula see the
Proof of Various Integral Properties section of the Extras chapter.
Let�s work a couple of quick examples.
Example
Determine the average value of each of the following functions on the
given interval.
(a)
 on
(b)
 on
Solution
(a)
 on
There�s really not a whole lot to do in this problem other
than just use the formula.
You caught the substitution needed for the third term
right?
So, the average value of this function of the given
interval is -1.620993.
(b)
 on
Again, not much to do here other than use the formula.
Note that the integral will need the following substitution.
Here is the average value of this function,
So, in this case the average function value is zero. Do
not get excited about getting zero here. It will happen on occasion. In fact,
if you look at the graph of the function on this interval it�s not too hard to
see that this is the correct answer.
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