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.
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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 \( \theta = \theta_1 \).
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
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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
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Practice Instructions
Unless otherwise instructed, calculate the arc length of these polar curves, giving your answers in exact, simplified form.