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### Section 9.9 : Arc Length with Polar Coordinates

1. Determine the length of the following polar curve. You may assume that the curve traces out exactly once for the given range of $$\theta$$.

$r = - 4\sin \theta , \,\, 0 \le \theta \le \pi$

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Start Solution

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

$\frac{{dr}}{{d\theta }} = - 4\cos \theta$ Show Step 2

We’ll need the $$ds$$ for this problem.

\begin{align*}ds & = \sqrt {{{\left[ { - 4\sin \theta } \right]}^2} + {{\left[ { - 4\cos \theta } \right]}^2}} \,d\theta \\ & = \sqrt {16{{\sin }^2}\theta + 16{{\cos }^2}\theta } \,d\theta = 4\sqrt {{{\sin }^2}\theta + {{\cos }^2}\theta } \,d\theta = 4d\theta \end{align*} Show Step 3

The integral for the arc length is then,

$L = \int_{{}}^{{}}{{ds}} = \int_{0}^{\pi }{{4\,d\theta }}$ Show Step 4

This is a really simple integral to compute. Here is the integral work,

$L = \int_{0}^{\pi }{{4\,d\theta }} = \left. {4\theta } \right|_0^\pi = \require{bbox} \bbox[2pt,border:1px solid black]{{4\pi }}$