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MathCalculus

Find the Exact Length of the Polar Curve r = 1 − sin θ on 0 ≤ θ ≤ 2π

Polar arc length is ∫√(r² + (dr/dθ)²) dθ. For r = 1 − sin θ the radicand reduces to 2 − 2 sin θ, a half-angle identity turns it into a sine, and the exact value of the integral is 8.

Question

For the following exercises, find the length of the curve over the given interval.

  1. r=1−sin⁡θr=1-\sin\theta on the interval 0≤θ≤2π0\le\theta\le 2\pi

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Evidence boundary

This page works the curve published as exercise 222 in the OpenStax Calculus Volume 2 §7.4 exercise set, whose instruction is to find the length of the curve over the given interval. The value 8 is the exact value of that integral rather than a decimal approximation, and the whole computation is carried out on the published interval 0 ≤ θ ≤ 2π. The page does not claim the problem comes from any other textbook, worksheet or answer key, and it does not extend the result to a different interval or to a different cardioid.