Introduction to Integration - The Exercise Bicycle Problem, Part 2
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.
- 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.
- Do you see how her method for estimating rate relates to the horizontal green lines?
- 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
- 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?
- Justify: Estimating the total distance by using shorter time intervals
(more divisions) produces more accurate estimates.
- 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.
- Similar to the previous problem, imagine we check the speed every second, and use that to
estimate total distance.
- How many terms will be in the estimate?
- Use `f` to write out the first three and final three terms of the estimate