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