Measuring Curved Paths

Calculus 1 · Axiom Academy

A ruler measures straight lines. So how long is a curve? Pull it straight, one tiny piece at a time. You can't lay a ruler along a bend Stretch a string along a winding road, then pull the string straight and measure it — that is the length of the road. Calculus does exactly this, but with infinitely many infinitely short straight pieces. Watch it happen first, then drive it yourself. A point walks along the curve, dropping a straight segment between each step. Each segment is a shortcut you can measure with a ruler. Add the segment lengths as they fall, and the running total climbs to the true length of the curve. The length of a curve is just the running total of the straight pieces — the limit of what a ruler can reach. Few big chords undershoot — so chop finer A handful of long chords cut across the bends and come up short. Slide the handle to chop the curve into more pieces: each chord hugs the curve tighter, the chunky polygon tightens onto the true path, and the total length climbs toward the real answer — the gap shrinking to zero. As Δx → 0 the polygon becomes the curve, and the total length becomes the exact length — that limit is the arc length. Each straight piece is a right triangle

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