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Section 9.9 : Arc Length with Polar Coordinates

For problems 1 – 3 determine the length of the given polar curve. For these problems you may assume that the curve traces out exactly once for the given range of \(\theta \).

1. \(\displaystyle r = \frac{1}{{\cos \theta }}\), \(\displaystyle 0 \le \theta \le \frac{\pi }{3}\)
2. \(r = {\theta ^2}\), \(0 \le \theta \le 3\pi \)
3. \(r = 6\cos \theta - 3\sin \theta \), \(0 \le \theta \le \pi \)

For problems 4 – 6 set up, but do not evaluate, an integral that gives the length of the given polar curve. For these problems you may assume that the curve traces out exactly once for the given range of \(\theta \).

1. \(r = \sin \left( {{\theta ^2}} \right)\), \(0 \le \theta \le \pi \)
2. \(r = \cos \left( {1 + \sin \theta } \right)\), \(0 \le \theta \le 2\pi \)
3. \(r = {{\bf{e}}^{ - \,\,\frac{1}{4}\theta }}\cos \theta \), \(0 \le \theta \le 3\pi \)