Length of a Curve
Calculus
Any curve, research the rule and show you understand it.
Introduction
Extending the bridge problem, how long is the curved cable on the bridge? Can you approximate it? Extension: Could you build this in GeoGebra to be dynamic, so that a user can choose a height and a length of the bridge?
Guiding Questions
- Pick a curve — a road bend, a slide, a function you like. What does 'length of the curve' mean, and how could you estimate it without a formula?
- Approximate the length by splitting the curve into many short straight segments and adding up their lengths with Pythagoras. What happens to your estimate as the segments get shorter?
- Check your answer with a completely different method: string laid along a printout, or a tape measure along a physical curve.
- How does the length change as you stretch or scale the curve? Find the rule.