Section 5.4
1. Find the average value of the function f (x) = x
2
2. Find the average value of the function f (x) = 1/x
on [0, 3]
2
on [1, 5]
3. Find the average value of the function f (x) = 36 - x
2
on [ - 2, 2].
4. Find the average value of the function f (x) = 3 on [10, 50].
5. Find the average value of the function f (x) = e
6. Find the average value of the function
0.01x
f (x)  e x
4
on [0, 10].
on [ - 1, 1].
7. BUSINESS: Sales – A store’s sales on day x are given by
4
S (x) = 200x + 6x . Find the average sales during the first three days.
8. ENVIRONMENTAL SCIENCE: Pollution – The amount of pollution in a lake x years
after the closing of a chemical plant is P (x) = 100/x tons. Find the average
amount of pollution between 1 and 10 years after closing.
9. BUSINESS: Average Income – Microsoft’s net income during the years 2003-2007
3
2
was approximately I (x) = - 16x + 72x – 60x + 90, hundred million dollars, where
x is the number of years since 2003. Find the average income during this period.
10. Find the area between the following two curves.
2
Y = 2x – 1 and y = x + 1 from x = 0 to x = 3.
11. Find the area between the following two curves.
2
Y = x + 4 and y = 2x + 1 from x = 0 to x = 3.
12. Find the area between the following two curves.
2
2
Y = x - 1 and y = 2 - 2x .
13. Find the area between the following two curves.
2
Y = 3x - 12 and y = 0.
14. Find the area between the following two curves.
2
3
Y = x and y = x .
0.02t
15. GENERAL : Population: The birth rate in Africa has increased from 17 e
to 22 e
0.02t
million births per year, where t is the number of years since 2000. find the total
increase in population that will result for the higher birth rate between 2000 and
2020.
16. BUSINESS: Net Savings – A factory installs new machinery that saves S (x) = 1200
– 20x dollars per year, where x is the number of years since installation. However,
the cost of maintaining the new machinery is C (x) = 100x dollars per year.
a. Find the year in which the maintenance cost will exceed the savings. (At this
time the new machinery should be replaced.)
b. Find the accumulated net savings during the period from t = 0 to the
replacement time period found in part a.
0.2t
17. ECONOMICS: Balance of Trade – A countries annual imports are I (t) = 30 e and
0.1t
its exports are E (t) = 25 e , both in billions of dollars, where t is measured in
years and t = 0 corresponds to the beginning of 2000. Find the countries
accumulated trade deficits for the 10 year period beginning in 2000.