James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 6.5 Average Value of a Function 461
It is easy to calculate the average value of finitely many numbers y 1 , y 2 , . . . , y n :
T
15
10
5
6
T ave
0 12 18 24 t
FIGURE 1
y ave − y 1 1 y 2 1 ∙ ∙ ∙ 1 y n
n
But how do we compute the average temperature during a day if infinitely many temperature
readings are possible? Figure 1 shows the graph of a temperature function Tstd,
where t is measured in hours and T in °C, and a guess at the average temperature, T ave .
In general, let’s try to compute the average value of a function y − f sxd, a < x < b.
We start by dividing the interval fa, bg into n equal subintervals, each with length
Dx − sb 2 adyn. Then we choose points x 1
*, . . . , x n * in successive subintervals and calculate
the average of the numbers f sx 1
*d, . . . , f sx n *d:
f sx 1
*d 1 ∙ ∙ ∙ 1 f sx n *d
n
(For example, if f represents a temperature function and n − 24, this means that we take
temperature readings every hour and then average them.) Since Dx − sb 2 adyn, we can
write n − sb 2 adyDx and the average value becomes
f sx 1
*d 1 ∙ ∙ ∙ 1 f sx n *d
b 2 a
Dx
− 1
b 2 a f f sx 1 * d 1 ∙ ∙ ∙ 1 f sx n *dg Dx
− 1
b 2 a f f sx 1 * d Dx 1 ∙ ∙ ∙ 1 f sx n *d Dxg
− 1
b 2 a on f sx i
*d Dx
i−1
If we let n increase, we would be computing the average value of a large number of
closely spaced values. (For example, we would be averaging temperature readings taken
every minute or even every second.) The limiting value is
lim
n l `
1
b 2 a on f sx*d i Dx − 1
i−1
b 2 a yb f sxd dx
a
For a positive function, we can think of
this definition as saying
area
− average height
width
by the definition of a definite integral.
Therefore we define the average value of f on the interval fa, bg as
f ave − 1
b 2 a yb f sxd dx
a
Example 1 Find the average value of the function f sxd − 1 1 x 2 on the
interval f21, 2g.
SOLUtion With a − 21 and b − 2 we have
f ave − 1
b 2 a yb f sxd dx −
a
1
2 2 s21d y2 s1 1 x 2 d dx − 1 Fx 1 x 3
21
3
2
3G21
− 2
n
If Tstd is the temperature at time t, we might wonder if there is a specific time when
the temperature is the same as the average temperature. For the temperature function
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.