310 STEP 4. Review the Knowledge You Need to Score High
13.1 Average Value of a Function
Main Concepts:Mean Value Theorem for Integrals, Average Value of a Function on [a,b]
Mean Value Theorem for Integrals
Iffis continuous on [a,b], then there exists a numbercin [a,b] such that
∫b
a
f(x)dx=
f(c)(b−a). (See Figure 13.1-1.)
0 acbx
y
(c, f(c))
f(x)
Figure 13.1-1
Example 1
Givenf(x)=
√
x−1, verify the hypotheses of the Mean Value Theorem for Integrals for
f on [1, 10] and find the value ofcas indicated in the theorem.
The functionfis continuous forx≥1, thus:
∫ 10
1
√
x− 1 dx=f(c)(10−1)
2(x−1)^1 /^2
3
] 10
1
= 9 f(c)
2
3
[
(10−1)^1 /^2 − 0
]
= 9 f(c)
18 = 9 f(c); 2=f(c); 2=
√
c−1; 4=c− 1
5 =c.