Arc Length in Polar

Calculus 2 · Axiom Academy

Build the polar arc-length integral straight from one tiny segment of the curve — then watch real curves measure themselves. 1. One Tiny Step Along the Curve Pick a point on a polar curve at angle , then nudge the angle by a hair to . The point moves a little. That tiny move splits into two perpendicular legs: a radial leg dr (the radius grew) and a transverse leg (the point swept sideways along an arc of radius r ). The step itself is the hypotenuse. radial leg — how much r changed transverse leg — arc of radius r over angle 2. Add the Steps — and Pull Out To get the whole length we sum ds along the curve. Since is the variable we integrate in, factor a out of the square root. Inside, becomes , and the comes out as a single : The picture below shows why this works. Replace the curve with straight chords between sampled points . Each chord is a real ds . Use more chords and the polyline hugs the cardioid — its measured length climbs toward the true arc length L = 8 . 3. The Integral Sweeps the Whole Curve Now let sweep from to . At each angle the integrand is the rate at which arc length accumulates per unit angle. Multiply by , add them up, and the running total is the arc length traced so far. The integrand: for the cardioid we have , so . The running total: sweeping , the live readout climbs and settles on the full length — exactly 8 for this cardioid. Here , so and the integral collapses to . Most polar curves are not this kind. 4. When the Square Root Fights Back

This is the written version of the interactive lesson above. See the full Calculus 2 course.