5 Steps to a 5 AP Calculus BC 2019

More Applications of Definite Integrals 311

Example 2

Givenf(x)=x^2 , verify the hypotheses of the Mean Value Theorem for Integrals for fon

[0, 6] and find the value ofcas indicated in the theorem.

Sincefis a polynomial, it is continuous and differentiable everywhere,

∫ 6

0

x^2 dx= f(c)(6−0)

x^3

3

] 6

0

= f(c)6

72 = 6 f(c); 12= f(c); 12=c^2

c=±

√

12 =± 2

√

3

(

± 2

√

3 ≈± 3. 4641

)

.

Since only 2

√

3 is in the interval [0, 6],c= 2

√

3.

TIP • Remember: if f′is decreasing, then f′′ < 0 and the graph of f is concave
downward.

Average Value of a Function on [a, b]

Average Value of a Function on an Interval

If f is a continuous function on [a,b], then the Average Value of f on [a,b]

=

1

b−a

∫b

a

f(x)dx.

Example 1

Find the average value ofy=sinxbetweenx=0 andx=π.

Average value=

1

π− 0

∫π

0

sinxdx

=

1

π

[−cosx]π 0 =

1

π

[−cosπ−(−cos(0))]

=

1

π

[1+1]=

2

π

.