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
1 b
yav 
f
(x)
dx.

ba a
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
x
dx 

20 0
2
2
1x 
2  3  0
3

1  23 03 
 

2 3
3

1 8

2 3

4
3
1
 1
3
Find the average value of the function
over the given interval:
f ( x)  e ; 0,4
x
f ( x)  4  x ;  2,3
2
A group of marine biologists estimates
that if certain conservation measures are
implemented, the population of an
endangered species of whale will be
N (t )  3t  2t  10t  600
3
2
0  t  10
where N(t) denotes the pop at the end of
year t. Find the average pop of the whales
over the next 10 years.
The sales of the USI company in the
first t years of its operation are
approximated by
S (t )  t 0.2t  4
2
where S(t) is measured in millions of
dollars. What were the average sales
over the first 5 years?
The cost function for a certain
product is C ( x)  50,000  350 x
(a) Find the average cost per unit for
500 units. C (x)
(b) Find the average value of the cost
function over the interval
[0, 500 ].
(c) Explain the difference in (a) & (b).