Length of Spiral r = eᶿ

Calculus 2 · Axiom Academy

Find the arc length of the logarithmic spiral from to using the polar arc length formula. The logarithmic spiral grows outward as it turns. Find the length of the arc it traces over one full revolution, , using the polar arc length formula . One revolution of r = e to the θ The curve starts near the pole at (where r = 1 , green dot) and spirals outward to (where , orange dot). Because each turn scales by the same factor, the shape is self-similar — the same spiral at every magnification. Nice work — you found the arc length of a logarithmic spiral over one full turn. Here's what carried the solution: Polar arc length formula: for a polar curve , the arc length is . Always start by finding . The spiral's special structure: since , the integrand collapses to — one clean factoring step turns a messy radical into something easy to integrate. Exponential integration: , so the whole integral is just . Result: evaluating from 0 to (and using e^ 0 = 1 ) gives . The logarithmic spiral keeps the same shape at every scale — a property that shows up in seashells, galaxies, and hurricanes. This calculation measures exactly how far the curve travels in a single revolution of that endlessly self-similar shape.

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