Finding Arc Length of r(t) = <3t, 4sin t, 4cos t> from t=0 to t=2π

Calculus 3 · Axiom Academy

Differentiate the position vector, find the constant speed, then integrate to get the arc length. The components and trace a radius-4 circle while 3t carries the point steadily along the x -axis — one full turn as t runs from 0 to . The speed turns out to be constant. Find the arc length of the curve traced by the vector function from t = 0 to . Nice work — you found the arc length of a space curve from start to finish. Here's what carried the solution: Arc length formula: for a vector function , the length is — the integral of the speed. Velocity vector: differentiate each component of to get . Pythagorean identity: collapsed the trig terms, leaving . Constant speed: because the speed was constant, the integral reduced to . The method works for any smooth curve: find , compute its magnitude, and integrate over the interval.

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