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17Calculus Polar Coordinates - Arc Length

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On this page we cover a common calculus problem involving polar coordinates, determining arc length.
As mentioned on the main polar coordinates page, polar coordinates are just parametric equations. If you are familiar with parametric equations, this material should be very intuitive.

To calculate the arc length of a smooth curve, we can use the integral \[ s = \int_{\theta_0}^{\theta_1}{\sqrt{r^2 + [r']^2} ~ d\theta} \] where r is a function of \(\theta\), \( r' = dr/d\theta \) and we are looking at the arc from \( \theta = \theta_0 \) to \( \theta = \theta_1 \).

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Practice

Unless otherwise instructed, calculate the arc length of these polar curves, giving your answers in exact, simplified form.

Basic

\( r=e^{\theta/2} \), \(0\leq\theta\leq4\pi\)

Problem Statement

Calculate the arc length of the polar curve \( r=e^{\theta/2} \), \(0\leq\theta\leq4\pi\).

Solution

1385 video solution

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Intermediate

\(r=\sin^2(\theta/2)\), \(0\leq\theta\leq\pi\)

Problem Statement

Calculate the arc length of the polar curve \(r=\sin^2(\theta/2)\), \(0\leq\theta\leq\pi\).

Solution

1386 video solution

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Practice Instructions

Unless otherwise instructed, calculate the arc length of these polar curves, giving your answers in exact, simplified form.

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