Arc Length of Curve in Polar Coordinates/Function of Angle

Theorem

Let $C$ be a curve embedded in a polar plane.

Let the radial coordinate $r$ of $C$ be defined as a function of the angular coordinate $\theta$:

$r = \map f \theta$

The arc length $s$ of $C$ between $\theta = \alpha$ and $\theta = \beta$ is defined as:

$\ds s := \int_\alpha^\beta \paren {\sqrt {\paren {\frac {\d r} {\d \theta} }^2 + r^2} } \rd \theta$

Proof

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Also see

• Arc Length of Curve in Polar Coordinates as Function of Radius

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): length
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): length