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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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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,

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