9.9-bo'lim : Qutb koordinatalari bilan yoy uzunligi

1-3-masalalar uchun berilgan qutbli egri chiziq uzunligini aniqlang. Ushbu muammolar uchun egri chiziq \(\theta \) ning berilgan diapazoni uchun aynan bir marta chiqadi deb taxmin qilishingiz mumkin.

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. \(r = 6\cos \theta - 3\sin \theta \), \(0 \le \theta \le \pi \)

4-6 muammolar uchun berilgan qutb egri chizig'ining uzunligini beradigan integralni o'rnating, lekin baholamang. Ushbu muammolar uchun siz egri chiziq \(\theta \) ning berilgan diapazoni uchun aynan bir marta chiqadi deb taxmin qilishingiz mumkin.

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