- Roger decides to run a marathon. Roger’s friend Jeff rides behind him on a bicycle and clocks his pace every 15 minutes. Roger starts out strong, but after an hour and a half he is so exhausted that he has to stop. The data Jeff collected are summarized below:

Time spent running (min) | 0 |
15 |
30 |
45 |
60 |
75 |
90 |

Speed (mph) | 12 |
11 |
10 |
10 |
8 |
7 |
0 |

- Assuming that Roger’s speed is never increasing, give upper and lower estimates for the distance Roger ran during the first half hour.
- Give upper and lower estimates for the distance Roger ran in total during the entire hour and a half.
- How often would Jeff have needed to measure Roger’s pace in order to find lower and upper estimates within 0.1 mile of the actual distance that he ran?

- Coal gas is produced at a gasworks. Pollutants in the gas are removed by scrubbers, which become less and less efficient as time goes on. The following measurements, are made at the start of each month, show the rate at which pollutants are escaping in the gas:

Time (months) | 0 |
1 |
2 |
3 |
4 |
5 |
6 |

Rate pollutants are escaping (tons/month) | 5 |
7 |
8 |
10 |
13 |
16 |
20 |

- Make an overestimate and an underestimate of the total quantity of pollutants that escaped during the first month.
- Make an overestimate and an underestimate of the total quantity of pollutants that escaped during the six months.
- How often would measurements have to be made in order to find overestimates and underestimates which differ by less than 1 ton from the exact quantity of pollutants that escaped during the first six months?

For problems 3-5, use Maple to calculate left- and right-hand sums and mid-point sums with 2, 10, 50, 250 subdivisions. Observe the limit to which your sums are tending as the number of subdivisions gets larger, and hence estimate the value of the definite integral.

- Under the curve for 0 £ x £ 2.
- Between y = x
^{1/2}and y = x^{1/3}for 0 £ x £ 1. - Consider the definite integral

For problems 6 and 7 estimate the area of the regions.

- Write an expression for a right-hand sum approximation for this integral using n subdivisions. Express each x
_{i}, i = 1, 2, ..., n, in terms of i.- Use a Maple to obtain a formula for the sum you wrote in part (a) in terms of n.
- Take the limit of this expression for the sum as n® ¥ , thereby finding the exact value of this integral.