Integrals
1.
The Internet is defined as a worldwide interconnection of individual networks operated by governments, industries, academia, and private parties (Internet World Stats, 2018).^{ }As the Internet grows and becomes more accessible to many people in the world, there is an increasing interest in forecasting its number of users because this helps improve the technology and its security.
In a recent study, researchers found that the rate of change of Internet users in the world (in millions) is modeled by the following function:
where x is the number of years from 1994.
For the questions that require calculations, be sure to show all of your work details. If applicable, round your final answers to the nearest 2 decimal places to receive full credit. For critical thinking questions, explain in detail how the answers were determined.
1.Using integration on, find the function P(x) that predicts the number of Internet users in the world years after 1994. (Note: Because this is an indefinite integral, do not forget to add a constant of integration, C, in your final answer.) Show all of your work details.
2.Based on the first letter of your last name, calculate the value of C from Question 1 by choosing an ordered pair (x, P(x)) from the table below. Show all of your work details.
First Letter of Your Last Name |
x (years from 1994) |
P(x) (in millions) |
A–B |
6 |
361 |
C–D |
7 |
513 |
E–F |
8 |
587 |
G–H |
9 |
719 |
I–J |
10 |
817 |
K–L |
11 |
1,018 |
M–N |
12 |
1,093 |
O–P |
13 |
1,319 |
Q–R |
14 |
1,574 |
S–T |
15 |
1,802 |
U–V |
16 |
2,267 |
W–X |
17 |
2,497 |
Y–Z |
18 |
2,802 |
3.From your answers in Questions 1 and 2, state the mathematical model P(x) (in millions) that will predict the number of Internet users in the world.
4.Using P(x), complete the following table. Show all of your work details for full credit.
Year |
x |
Predicted Number of Internet Users (in millions) |
2016 |
22 |
2,526.28 |
2018 |
24 |
3,020.86 |
2020 |
26 |
3,578 |
5.Below is a graph of over the interval [6, 22]. Set up the definite integral that will calculate the area of the shaded region on the graph.
6.Calculate the area of the shaded region. Show all of your work details.
7.What information does the area give in terms of the number of Internet users in this scenario? Would you estimate that this is a reasonable growth of Internet users from 2000 to 2016? Explain and justify your answer with critical thinking.
2.
To find the area under a curve between two points, you can use integration. Given the following two functions, answer the questions:
a.What is the best method used to find the highlighted area in the graph? What can you tell about this method?
- In this example, you can see how you can find an area using integrals. How can integration be used to find a volume?
- Describe 2 examples in which integration is used in real-world applications.
References
Internet World Stats. (2018, April 3). Internet growth statistics. Retrieved from http://www.internetworldstats.com/emarketing.htm
SPEED OF ME. (2018). SpeedOf.Me. Retrieved from http:www.speedof.me/