Arc Length (Parametric)

AP Calculus · Axiom Academy

LESSON Arc Length for Parametric Curves Finding the distance traveled along a curved path Imagine walking along a parametric curve. How far have you traveled? This is arc length! For a parametric curve: We need to sum up all the tiny distances traveled as t goes from a to b. The key insight: as dt changes by a small amount, the distance traveled is approximately: This comes from the Pythagorean theorem! Each small segment has horizontal change dx and vertical change dy. where the curve is traced once as t goes from a to b This is the fundamental formula! It says: integrate the speed (magnitude of velocity vector) from start to finish. Find the arc length of the circle: from t = 0 to t = 2π. Step 2: Calculate (dx/dt)² + (dy/dt)² Step 4: Integrate from 0 to 2π This is exactly the circumference of a circle with radius 1! Beautiful! Not all parametric arc length integrals are easy to evaluate. Here are some strategies:

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