Arc Length of Space Curves

Calculus 3 · Axiom Academy

LESSON Arc Length of Space Curves How to measure the length of a curve in three-dimensional space Imagine walking along a curved path in 3D space. A ruler is straight, so to measure the total distance you have travelled we break the path into many tiny straight-line segments (chords), measure each one with the 3D distance formula, and add them up. One chord: the 3D distance between two nearby points The exact length is the limit of those sums Consider a parametric curve for . Over a small interval , the change in position is approximately the velocity times the elapsed parameter: The displacement vector over a small The length of that small displacement is found with the 3D distance formula — it is the space diagonal of the little box whose edges are , and : , and are the components of — the legs of the box. is the chord joining the two points. The chord swings onto the tangent direction: . . For the helix that limit is . Taking the limit as turns each chord into , and summing them turns the sum into an integral. Every chord contributes its own length, so the total is an accumulation. What is being accumulated is the speed — the rate at which arc length is produced per unit of t . Or equivalently, using the magnitude of the velocity vector: Here is the velocity vector and is its magnitude — the speed. is the height of the graph at each t . The area swept out under that graph up to t is exactly the distance travelled so far.

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