Calculus II - Arc Length with Polar Coordinates

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Section 9.9 : Arc Length with Polar Coordinates

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2. Set up, but do not evaluate, and integral that gives the length of the following polar curve. You may assume that the curve traces out exactly once for the given range of $\theta$ .

$r = \theta \cos \theta , \,\,0 \le \theta \le \pi$

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The first thing we’ll need here is the following derivative.

$\frac{{dr}}{{d\theta }} = \cos \theta - \theta \sin \theta$ Show Step 2

We’ll need the $ds$ for this problem.

$ds = \sqrt {{{\left[ {\theta \cos \theta } \right]}^2} + {{\left[ {\cos \theta - \theta \sin \theta } \right]}^2}} \,d\theta$ Show Step 3

The integral for the arc length is then,

$\require{bbox} \bbox[2pt,border:1px solid black]{{L = \int_{0}^{\pi }{{\sqrt {{{\left[ {\theta \cos \theta } \right]}^2} + {{\left[ {\cos \theta - \theta \sin \theta } \right]}^2}} \,d\theta }}}}$