Introduction to Integration - The Exercise Bicycle Problem, Part 1


An aerobics instructor is designing a 1-hour workout for the stationary bicycle. For best results, the speed of the wheels should stay between 10 mph (miles per hour) and 15 mph. You can see five different workouts she is considering in the applet above. For instance, in example ride 1 she will spend the first 30 minutes of her ride at 10 mph, and the last 30 minutes of the ride at 15 mph.

The big question: How far has she traveled? Ok, ok, yes, the real answer is 0 miles since this is a stationary bicycle! But what is a good estimate for "virtual distance" traveled?


  1. What equation relates distance, rate, and time?
  2. Which example results in the greatest distance traveled? Look through the examples first, and see if you can intuitively guess, then compute the distance for each example.
  3. What is the relationship between distance traveled and the area under the graph? (In the applet you can shade the area under the graph to see it better.) Explain why this relationship holds.
  4. In "real life," stationary bicycles often have a small electronic display mounted on the handlebars that shows speed and "distance". How do you think the distance is calculated? Consider the fact that a rider's speed can change quite quickly.