9.9-bo'lim : Qutb koordinatalari bilan yoy uzunligi
1. Quyidagi qutbli egri chiziq uzunligini aniqlang. Egri chiziq \(\theta \) ning berilgan diapazoni uchun aynan bir marta chiqadi deb taxmin qilishingiz mumkin.
\[r = - 4\sin \theta , \,\, 0 \le \theta \le \pi \]Yechim
Bu erda bizga kerak bo'lgan birinchi narsa - bu quyidagi lotin.
\[\frac{{dr}}{{d\theta }} = - 4\cos \theta \]2-qadam
Ushbu muammo uchun bizga \(ds\) kerak bo'ladi.
\[\begin{align*}ds & = \sqrt {{{\left[ { - 4\sin \theta } \right]}^2} + {{\left[ { - 4\cos \theta } \right]}^2}} \,d\theta \\ & = \sqrt {16{{\sin }^2}\theta + 16{{\cos }^2}\theta } \,d\theta = 4\sqrt {{{\sin }^2}\theta + {{\cos }^2}\theta } \,d\theta = 4d\theta \end{align*}\]3-qadam
U holda yoy uzunligi uchun integrali,
\[L = \int_{{}}^{{}}{{ds}} = \int_{0}^{\pi }{{4\,d\theta }}\]4-qadam
Bu hisoblash uchun juda oddiy integraldir. Mana integral ish,
\[L = \int_{0}^{\pi }{{4\,d\theta }} = \left. {4\theta } \right|_0^\pi = \require{bbox} \bbox[2pt,border:1px solid black]{{4\pi }}\]