16.7-bo'lim : Grin teoremasi
1. \( \displaystyle \int\limits_{C}{{y{x^2}\,dx - {x^2}\,dy}}\) ni baholash uchun Grin teoremasidan foydalaning, bu erda \(C\) quyida ko'rsatilgan.
Yechim
Xo'sh, birinchi navbatda e'tibor bering, agar biz yo'l bo'ylab ko'rsatilgan yo'nalishda yuradigan bo'lsak, chap qo'limiz yopiq maydon ustida bo'ladi va shuning uchun bu yo'l ijobiy yo'nalishga ega va biz integralni baholash uchun Grin teoremasidan foydalanishimiz mumkin.
Bizda mavjud integraldan,
\[P = y{x^2}\hspace{0.25in}\hspace{0.25in}Q = - {x^2}\]Esda tutingki, \(P\) \(x\) ga, \(Q\) esa \(y\) ga ko'paytiriladi va belgilarga e'tibor berishni unutmang. Shoshilinch va shartlardan birining oldidagi belgini o'tkazib yuborish oson.
2-qadam
Green teoremasidan foydalanib, chiziqli integralga aylanadi,
\[\int\limits_{C}{{y{x^2}\,dx - {x^2}\,dy}} = \iint\limits_{D}{{ - 2x - {x^2}\,dA}}\]\(D\) - egri chiziq bilan o'ralgan hudud.
3-qadam
\(D\) faqat yarim doira bo'lgani uchun bu muammo uchun qutb koordinatalaridan foydalanish mantiqan. Qutb koordinatalarida \(D\) chegaralari:
\[\begin{array}{c}\displaystyle \frac{1}{2}\pi \le \theta \le \frac{3}{2}\pi \\ 0 \le r \le 5\end{array}\]4-qadam
Endi biz qilishimiz kerak bo'lgan narsa, albatta, qutb koordinatalariga aylantirilgandan so'ng, ikki karrali integralni baholashdir.
Bu erda baholash ishi.
\[\begin{align*}\int\limits_{C}{{y{x^2}\,dx - {x^2}\,dy}} & = \iint\limits_{D}{{ - 2x - {x^2}\,dA}}\\ & = \int_{{\frac{1}{2}\pi }}^{{\frac{3}{2}\pi }}{{\int_{0}^{5}{{r\left( { - 2r\cos \theta - {r^2}{{\cos }^2}\theta } \right)dr}}\,d\theta }}\\ & = \int_{{\frac{1}{2}\pi }}^{{\frac{3}{2}\pi }}{{\int_{0}^{5}{{ - 2{r^2}\cos \theta - \frac{1}{2}{r^3}\left( {1 + \cos \left( {2\theta } \right)} \right)dr}}\,d\theta }}\\ & = \int_{{\frac{1}{2}\pi }}^{{\frac{3}{2}\pi }}{{\left. {\left( { - \frac{2}{3}{r^3}\cos \theta - \frac{1}{8}{r^4}\left( {1 + \cos \left( {2\theta } \right)} \right)} \right)} \right|_0^5\,d\theta }}\\ & = \int_{{\frac{1}{2}\pi }}^{{\frac{3}{2}\pi }}{{ - \frac{{250}}{3}\cos \theta - \frac{{625}}{8}\left( {1 + \cos \left( {2\theta } \right)} \right)\,d\theta }}\\ & = \left. {\left[ { - \frac{{250}}{3}\sin \theta - \frac{{625}}{8}\left( {\theta + \frac{1}{2}\sin \left( {2\theta } \right)} \right)} \right]} \right|_{\frac{1}{2}\pi }^{\frac{3}{2}\pi }\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{500}}{3} - \frac{{625}}{8}\pi = - 78.7703}}\end{align*}\]\(A\)-ni qutbli koordinatalarga aylantirishdan qo'shimcha \(r\)-ni unutmang va integrallarda ko'rsatiladigan ko'plab trig funktsiyalari bilan shug'ullanishimiz kerak bo'lgan barcha turli tirg identifikatorlarini eslab qolishingiz kerak.