Introduction to Integration - The Exercise Bicycle Problem, Part 2

HELP

An aerobics instructor has decided that the best 1-hour stationary bicycle workout occurs when the speed of the bicycle follows the gray curve shown above. If the speed of the bicycle follows the curve, what will be the total (virtual) distance traveled? We will estimate the distance.

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1. Check the "Show divisions" checkbox, and set the number of divisions to 6. Imagine that the instructor does the workout, following the speed indicated by the gray curve. Every 10 minutes she notes her speed, and figures that she was probably going approximately that fast during the previous 10 minutes.
1. Do you see how her method for estimating rate relates to the horizontal green lines?
2. If she estimates her distance this way, what is her estimate for total distance after 60 minutes? You can just look at the graph to estimate `y`-values.
2. The instructor does the workout again, following the speed indicated by the gray curve. This time, she checks her speed every 5 minutes, and figures that she was probably going approximately that fast during the previous 5 minutes. What will be her estimate for total distance traveled during the 1-hour workout?
3. Justify: Estimating the total distance by using shorter time intervals (more divisions) produces more accurate estimates.
4. Suppose the gray speed curve is given by the function `y = f(t)` where `t` is in hours (careful!). Consider the arithmetic needed to find an estimate for total distance if you checked your bicycle speed every minute (`1/60` of an hour). Your estimate would look like a sum with 60 terms. Using the symbolic notation `f` and not using the graph to estimate `y`-values, write out the first three and the final three terms of your estimate.
5. Similar to the previous problem, imagine we check the speed every second, and use that to estimate total distance.
1. How many terms will be in the estimate?
2. Use `f` to write out the first three and final three terms of the estimate