8.2 Volume and Average Value

DEFINITION:

Let f be a continuous function over a closed interval [a, b]. Its average value, yav, over [a, b] is given by

yav 1

b af (x) dx.

a

b

Example: Find the average value Of f(x) = x2 over the interval [0, 2].

Example: Find the average value of f(x) = x2 over the interval [0, 2].

1

2 0x2dx

0

2

1

2

x3

3

0

2

1

2

23

3

03

3

1

28

3

4

3 1

1

3

Find the average value of the function over the given interval:

3,2;4)( 2 xxf

4,0;)( xexf

A group of marine biologists estimates that if certain conservation measures are implemented, the population of an endangered species of whale will be

where N(t) denotes the pop at the end of year t. Find the average pop of the whales over the next 10 years.

100 6001023)( 23 tttttN

The sales of the USI company in the first t years of its operation are

approximated by

where S(t) is measured in millions of dollars. What were the average sales

over the first 5 years?

42.0)( 2 tttS

The cost function for a certain product is

(a) Find the average cost per unit for 500 units.

xxC 350000,50)(

(b) Find the average value of the cost function over the interval [0, 500 ].

)(xC

(c) Explain the difference in (a) & (b).