Mistake Master · AP Calculus · Unit 9 · Step-Through Animation
Chords Close On It. Staircases Never Do.
You'll learnto assemble the parametric arc length integrand from the same Pythagorean triangle Unit 8 used, to see why no 1 appears under the root once a parameter supplies both coordinates, 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 horizontal and vertical 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.
8 STEPS · 6 QUICK CHECKS · CHORDS CONVERGE, STAIRCASES DO NOT · v1
no 1 under this root · the limits are t-values
Before you start
What you're looking at
One plane, drawn at equal scale on both axes, so a length on screen is a length. The curve is x = 3t², y = 2t³ for t from 0 to 1 — it runs from the origin to (3, 2) and bends only gently.
The question
How long is that curve? The answer is defined as a limit: chop it into pieces, join the ends of each piece with a straight line, add those lines up, and take more and more pieces.
Watch for
Two chains are refined side by side, at the same n. One is made of chords and one of horizontal and vertical legs. Both close on the curve. Watch what happens to the two printed totals — only one of them moves.
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