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?