Chapter 5 | Integration
551
Figure 5.26 By the Mean Value Theorem, the continuous function f ( x ) takes on its average value at c at least once over a closed interval.
Find the average value of the function f ( x ) = x 2
over the interval ⎡
⎤ ⎦ and find c such that f ( c )
5.14
⎣ 0, 6
equals the average value of the function over [0, 6].
Example 5.16 Finding the Point Where a Function Takes on Its Average Value
3 x 2 dx =9, find c such that f ( c ) equals the average value of f ( x ) = x 2 over [0, 3].
Given ∫
0
Solution We are looking for the value of c such that
3
3−0 ∫
x 2 dx = 1 3
f ( c ) = 1
(9) =3.
0
Replacing f ( c ) with c 2 , we have
c 2 = 3 c = ± 3. Since − 3 is outside the interval, take only the positive value. Thus, c = 3 ( Figure 5.27 ).
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