9.9-bo'lim : Qutb koordinatalari bilan yoy uzunligi
2. O'rnating, lekin baholamang va quyidagi qutb egri chizig'ining uzunligini beradigan integral. \(\theta \) ning berilgan diapazoni uchun egri chiziq aynan bir marta chiqadi deb taxmin qilishingiz mumkin.
\[r = \theta \cos \theta , \,\,0 \le \theta \le \pi \]Yechim
Bu erda bizga kerak bo'lgan birinchi narsa - bu quyidagi lotin.
\[\frac{{dr}}{{d\theta }} = \cos \theta - \theta \sin \theta \]2-qadam
Ushbu muammo uchun bizga \(ds\) kerak bo'ladi.
\[ds = \sqrt {{{\left[ {\theta \cos \theta } \right]}^2} + {{\left[ {\cos \theta - \theta \sin \theta } \right]}^2}} \,d\theta \]3-qadam
U holda yoy uzunligi uchun integrali,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{L = \int_{0}^{\pi }{{\sqrt {{{\left[ {\theta \cos \theta } \right]}^2} + {{\left[ {\cos \theta - \theta \sin \theta } \right]}^2}} \,d\theta }}}}\]