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Chapter 5 | Integration
Dividing by b − a gives us
b − a ∫ a b
f ( x ) dx ≤ M .
m ≤ 1
Since 1 b − a ∫ a b f ( x ) dx is a number between m and M , and since f ( x ) is continuous and assumes the values m and M over ⎡ ⎣ a , b ⎤ ⎦ , by the Intermediate Value Theorem (see Continuity ), there is a number c over ⎡ ⎣ a , b ⎤ ⎦ such that
b − a ∫ a b
f ( c ) = 1
f ( x ) dx ,
and the proof is complete. □ Example 5.15 Finding the Average Value of a Function
Find the average value of the function f ( x ) =8−2 x over the interval [0, 4] and find c such that f ( c ) equals the average value of the function over [0, 4].
Solution The formula states the mean value of f ( x ) is given by 1 4−0 ∫ 0 4
(8−2 x ) dx .
We can see in Figure 5.26 that the function represents a straight line and forms a right triangle bounded by the x - and y -axes. The area of the triangle is A = 1 2 (base) ⎛ ⎝ height ⎞ ⎠ . We have A = 1 2 (4)(8) = 16. The average value is found by multiplying the area by 1/(4−0). Thus, the average value of the function is
1 4
(16) =4.
Set the average value equal to f ( c ) and solve for c .
8−2 c = 4 c = 2
At c =2, f (2) =4.
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