Section 4.5
1. Find the average value of the function f (x) = x 2 on [0, 3]
1
Use the definition for average value,
ba

b
a
f (x) dx, and your calculator.
Don’t forget the division by b – a or in this case 3.
2. Find the average value of the function f (x) = 1/x 2 on [1, 5]
Use the definition for average value,
1
ba

b
a
f (x) dx, and your calculator.
Don’t forget the division by b – a or in this case 4.
3. Find the average value of the function f (x) = 36 - x 2 on [ - 2, 2].
Use the definition for average value,
1
ba

b
a
f (x) dx, and your calculator.
4. Find the average value of the function f (x) = 3 on [10, 50].
Use the definition for average value,
1
ba

b
a
f (x) dx, and your calculator.
5. Find the average value of the function f (x) = e
Use the definition for average value,
1
ba

b
a
0.01x
on [0, 10].
f (x) dx, and your calculator.
6. Find the average value of the function f (x)  e x
4
Use the definition for average value,
1
ba

b
a
on [ - 1, 1].
f (x) dx, and your calculator.
7. BUSINESS: Sales – A store’s sales on day x are given by
S (x) = 200x + 6x 2 . Find the average sales during the first three days.

3
0
200x  6x2 dx
To find the average sales during the first 3 days, calculate the average value of S(x)
for x = 0 to x = 3.
And to get the average sales we will divide 954 by 3 = $318
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.
Average pollution = 1
9

10
1
100= 25.58 tons.
dx
x
9. BUSINESS: Average Income – Microsoft’s net income during the years
2003-2007 was approximately I (x) = - 16x 3 + 72x 2 – 60x + 90, hundred
million dollars, where x is the number of years since 2003. Find the average
income during this period.
Average income =
1
4

4
0
 16x3  72x2  60x  90 dx = 610 hundred million.
Microsoft’s average income during this period was $98 hundred million.
10. Find the area between the following two curves.
Y = 2x – 1 and y = x 2 + 1 from x = 0 to x = 3.

3
0
(2x  1)  (x2  1) dx
Area = 6.
11. Find the area between the following two curves.
Y = x 2 + 4 and y = 2x + 1 from x = 0 to x = 3.

3
0
(x2  4)  (2x  1) dx
Area = 9.
12. Find the area between the following two curves.
Y = x 2 - 1 and y = 2 - 2x 2 .

1
1
(x2  1)  (2  2x2 ) dx  4
Area = 4.
13. Find the area between the following two curves.
Y = 3x 2 - 12 and y = 0.

2
2
3x 2  12 dx
Area = 32.
14. Find the area between the following two curves.
Y = x 2 and y = x 3 .

1
0
x2  x3 dx
Area = 0.08333.
15. GENERAL : Population: The birth rate in Africa has increased from 17 e 0.02t 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.
To find the total increase in population we will integrate the difference of the two
functions from 0 to 20.

20
0
17e 0.02t  22e0.02t dt
total increase in population from 2000 to 2020 will be about 123 million.
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.
a. To find the point where maintenance
cost will exceed savings we will graph
both functions and find their point of
intersection..
a. To find the accumulated net savings we will
integrate the difference of the two functions from
0 to 10.

10
0
(120  20x)  (100x) dx  $6,000
17. ECONOMICS: Balance of Trade – A countries annual imports are I (t) =
30 e0.2t and its exports are E (t) = 25 e 0.1t, 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.
The accumulated trade deficit will be the imports minus the exports. We will
integrate that difference from 0 to 10.

10
0
30e 0.2t  25e0.1t dt
The accumulated trade deficit will be about 529 billion dollars.