Arc Length Formula

Calculus 2 · Axiom Academy

From the Pythagorean theorem to integration — measuring the exact length of a curve by summing infinitely many tiny hypotenuses. Consider a smooth curve y = f(x) running from x = a to x = b . How long is it? Unlike a straight line, we can't just use the distance formula once. But we can approximate the curve with a chain of straight chords — and as the chords get shorter, the chain hugs the curve and its total length closes in on the true arc length. Zoom in on a tiny piece of the curve between x and x + dx . Over that sliver the curve is, for all practical purposes, straight. The horizontal change is dx and the vertical change is dy ; the length of the little segment is what we'll call dL , the differential of arc length. 3. Applying the Pythagorean Theorem That infinitesimal segment is the hypotenuse of a right triangle whose legs are dx (horizontal) and dy (vertical). The right angle sits where the two legs meet, directly below the curve. This is the geometric heart of the whole formula — every tiny piece of the curve is the hypotenuse of its own little right triangle. 4. Expressing dy in Terms of dx Since y = f(x) , the derivative tells us how y responds to a nudge in x : . The slope of the tangent line at a point is the rise per unit run, so over a run of dx the rise is . Substituting that into the Pythagorean relation lets us write everything in terms of dx alone. Start with (dL)^2 = (dx)^2 + (dy)^2 and substitute :

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