9.10-bo'lim : Polar koordinatali sirt maydoni

1-4 muammolar uchun egri chiziqning sirt maydonini berilgan o'q atrofida aylantiradigan integral o'rnating, lekin baholamang. Ushbu muammolar uchun egri chiziq \(\theta \) ning berilgan diapazoni uchun aynan bir marta chiqadi deb taxmin qilishingiz mumkin.

1. \(r = \cos \left( {{{\bf{e}}^{ - \,\,\frac{1}{4}\theta }}} \right)\), \(\displaystyle 0 \le \theta \le \frac{\pi }{2}\) \(x\) o'qi atrofida aylantirildi.
2. \(r = \theta \sin \theta \), \(\displaystyle 0 \le \theta \le \frac{\pi }{2}\) \(y\) o'qi atrofida aylantirildi.
3. \(r = \cos \left( \theta \right)\sin \left( {2\theta } \right)\), \(\displaystyle 0 \le \theta \le \frac{\pi }{6}\) \(x\) o'qi atrofida aylantirildi.
4. \(r = \theta + \sin \theta \), \(\displaystyle \frac{\pi }{2} \le \theta \le \pi \) \(y\) o'qi atrofida aylantirildi.