542
Chapter 5 | Integration
∑ i =1 n
f ⎛
⎞ ⎠
⎝ x i *
⎞ ⎠ ∑ i =1 n ⎞ ⎠ ∑ i =1 n
⎛ ⎝ Δ x
f ⎛
⎞ ⎠
=
⎝ x i *
b − a
( b − a ) Δ x
⎛ ⎝ 1
f ⎛
⎞ ⎠ Δ x .
=
⎝ x i *
b − a
This is a Riemann sum. Then, to get the exact average value, take the limit as n goes to infinity. Thus, the average value of a function is given by 1 b − a lim n →∞ ∑ i =1 n f ( x i )Δ x = 1 b − a ∫ a b f ( x ) dx .
Definition Let f ( x ) be continuous over the interval ⎡ ⎣ a , b ⎤
⎦ . Then, the average value of the function f ( x ) (or f ave ) on ⎡ ⎣ a , b ⎤ ⎦ is
given by
b − a ∫ a b
f ( x ) dx .
f ave = 1
Example 5.14 Finding the Average Value of a Linear Function
Find the average value of f ( x ) = x +1 over the interval ⎡ ⎣ 0, 5 ⎤ ⎦ .
Solution First, graph the function on the stated interval, as shown in Figure 5.25 .
Figure 5.25 The graph shows the area under the function f ( x ) = x +1 over ⎡ ⎣ 0, 5 ⎤ ⎦ .
h ( a + b ), where
The region is a trapezoid lying on its side, so we can use the area formula for a trapezoid A = 1 2
h represents height, and a and b represent the two parallel sides. Then,
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