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Section 9.10 : Surface Area with Polar Coordinates

For problems 1 – 4 set up, but do not evaluate, an integral that gives the surface area of the curve rotated about the given axis. For these problems you may assume that the curve traces out exactly once for the given range of \(\theta \).

1. \(r = \cos \left( {{{\bf{e}}^{ - \,\,\frac{1}{4}\theta }}} \right)\), \(\displaystyle 0 \le \theta \le \frac{\pi }{2}\) rotated about the \(x\)-axis.
2. \(r = \theta \sin \theta \), \(\displaystyle 0 \le \theta \le \frac{\pi }{2}\) rotated about the \(y\)-axis.
3. \(r = \cos \left( \theta \right)\sin \left( {2\theta } \right)\), \(\displaystyle 0 \le \theta \le \frac{\pi }{6}\) rotated about the \(x\)-axis.
4. \(r = \theta + \sin \theta \), \(\displaystyle \frac{\pi }{2} \le \theta \le \pi \) rotated about the \(y\)-axis.