Section 4.5

1. Find the average value of the function f (x) = x 2 on [0, 3]

b

a

1Use the definition for average value, f (x) dx, and your calculator.

b a

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]

b

a

1Use the definition for average value, f (x) dx, and your calculator.

b a

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].

b

a

1Use the definition for average value, f (x) dx, and your calculator.

b a

4. Find the average value of the function f (x) = 3 on [10, 50].

b

a

1Use the definition for average value, f (x) dx, and your calculator.

b a

5. Find the average value of the function f (x) = e 0.01x on [0, 10].

b

a

1Use the definition for average value, f (x) dx, and your calculator.

b a

6. Find the average value of the function on [ - 1, 1].4xf (x) e

b

a

1Use the definition for average value, f (x) dx, and your calculator.

b a

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 2

0200x 6x dx

And to get the average sales we will divide 954 by 3 = $318

To find the average sales during the first 3 days, calculate the average value of S(x) for x = 0 to x = 3.

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.

10

1

1 100dx

9 xAverage pollution = = 25.58 tons.

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 = = 610 hundred million.4 3 2

0

116x 72x 60x 90 dx

4

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 2

0(2x 1) (x 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 2

0(x 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 2 2

1(x 1) (2 2x ) dx 4

Area = 4.

13. Find the area between the following two curves.Y = 3x 2 - 12 and y = 0.

2 2

23x 12 dx

Area = 32.

14. Find the area between the following two curves.Y = x 2 and y = x 3 .

1 2 3

0x x 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 0.02t 0.02t

017e 22e 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 0.2t 0.1t

030e 25e dt

The accumulated trade deficit will be about 529 billion dollars.