Arc Length of Curve in Polar Coordinates/Function of Radius

Theorem

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

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

$\theta = \map f r$

The arc length $s$ of $C$ between $r = u$ and $r = v$ is defined as:

$\ds s := \int_u^v \paren {\sqrt {1 + r^2 \paren {\frac {\d \theta} {\d r} }^2} } \rd r$

Proof

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Also see

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

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