Parametric Arc Length

Calculus 2 · Axiom Academy

When a point moves as , its speed is the rate arc length piles up — so integrating speed measures the curve. 1. From Velocity to Arc Length Watch a point ride a parametric curve . The velocity vector is tangent to the path — it points the way the point is heading, and its length is the point's speed . Velocity = the rate of change of position Speed = the length of that velocity vector 2. The Parametric Arc Length Formula Chop the path into tiny pieces. Each piece is almost a straight segment, and the straight-line distances inscribe the curve. Sum them, then let the pieces shrink: the sum becomes an integral of speed from t=a to t=b . This measures the distance traveled along the curve as t runs from a to b . Test the formula on a circle of radius r , parameterized by the angle t . As the point sweeps once around, the arc it leaves behind should total the circumference. A cycloid is the curve traced by a point on the rim of a circle of radius r as it rolls along a straight line. Its parametric form makes a once-hopeless length a clean integral. You've seen arc length as accumulated speed — and watched it deliver both a circle's circumference and a cycloid's surprising 8r . Scroll up to revisit any step.

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