5. Integration

# Chapter 5 Review Exercises

True or False. Justify your answer with a proof or a counterexample. Assume all functions and are continuous over their domains.

1. If for all then the right-hand rule underestimates the integral Use a graph to justify your answer.

#### Solution

False

2. 3. If for all then #### Solution

True

4. All continuous functions have an antiderivative.

Evaluate the Riemann sums for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.

5. over #### Solution exact answer: 4

6. over 7. over #### Solution exact answer: 5.870

8. over Evaluate the following integrals.

9. #### Solution 10. 11. #### Solution

1

12. Find the antiderivative.

13. #### Solution 14. 15. #### Solution 16. Find the derivative.

17. #### Solution 18. 19. #### Solution 20. The following problems consider the historic average cost per gigabyte of RAM on a computer.

Year 5-Year Change ($) 1980 0 1985 −5,468,750 1990 755,495 1995 −73,005 2000 −29,768 2005 −918 2010 −177 21. If the average cost per gigabyte of RAM in 2010 is$12, find the average cost per gigabyte of RAM in 1980.

#### Solution

$6,328,113 22. The average cost per gigabyte of RAM can be approximated by the function where is measured in years since 1980, and is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.

23. Find the average cost of 1GB RAM for 2005 to 2010.

#### Solution

\$73.36

24. The velocity of a bullet from a rifle can be approximated by where is seconds after the shot and is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: What is the total distance the bullet travels in 0.5 sec?

25. What is the average velocity of the bullet for the first half-second?

#### Solution  