Chords Close On It. Staircases Never Do.

You'll learnto build the parametric arc length integrand out of one right triangle, to bound any answer between the chord and the staircase before evaluating anything, to integrate over t-values rather than x-values, and to tell the length of a curve from the distance a particle travels along it.

Arc length is defined as a limit of summed chords, and this animation runs that limit rather than describing it. Beside it runs the commonest wrong version: a staircase of flat and upright legs, refined at exactly the same rate. The chords close on the curve and their total climbs to the answer. The staircase closes on the curve too, and its total does not move at all. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · CHORDS CONVERGE, STAIRCASES DO NOT · v2

square both rates, add them, and take the root of the sum

Before you start

What you're looking at

One plane at equal scale, so a length on screen is a length. The curve is x = 3t², y = 2t³ for t from 0 to 1, running from the origin to (3, 2).

The question

How long is that curve? Chop it up, join each piece end to end with a straight line, add those lines, and take more and more pieces.

Step 1 / 12