Arc Length of Parametric Curves

Calculus 2 · Axiom Academy

LESSON Arc Length of Parametric Curves How far does a point travel along a curve? Chop the path into tiny Pythagorean steps, add them up, and let them flow into an integral. 1. One Tiny Step Is a Right Triangle Take a point moving along the curve . Over a small change it slides from one spot to the next. Zoom in and that little hop is the hypotenuse of a right triangle with legs and — so its length is . The length of one straight hop along the curve 2. Add the Steps, Then Take the Limit One step is an approximation; the whole path is a chain of them. String n chords end-to-end along the curve and add their lengths. As you use more, shorter segments the polyline hugs the curve, and its total length climbs toward the true arc length L . Each chord has length . Factor out from under the root: . As the difference quotients become derivatives and the sum becomes . The chords shortcut across the bends, so the estimate sits below the true length. Shorter chords lie almost on the curve; the running total converges to L . Read the formula physically. The velocity vector is , and the thing under the root is exactly its magnitude — the speed of the particle. So arc length is just speed integrated over time : distance = rate time, summed up as the rate changes. points along the path, tangent to the curve. A long velocity arrow covers more arc per unit t — more length piles up. adds up every bit of path — even if the curve loops or backtracks. 4. Worked Example: A Circle of Radius 3

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