16.3-bo'lim : Chiziqli integrallar - II qism
6. Quyidagi egri chiziqlarning har biri uchun \( \displaystyle \int\limits_{C}{{\left( {x - y} \right)\,dx - y{x^2}\,dy}}\) ni baholang.
- \(C\) - radiusi 6 bo'lgan aylananing 1 dagi qismist, 2nd va 3rd soat yo'nalishi bo'yicha aylanish bilan kvadrant.
- \(C\) - \(\left( {0, - 6} \right)\) dan \(\left( {6,0} \right)\) gacha bo'lgan chiziq segmenti.
Yechim
Keling, muammoning ushbu qismi uchun egri chiziqning qisqacha chizmasidan boshlaylik.
Mana bu egri chiziq uchun parametrlash.
\[C:\,\,\vec r\left( t \right) = \left\langle {6\cos \left( t \right), - 6\sin \left( t \right)} \right\rangle \hspace{0.25in} \frac{1}{2}\pi \le t \le 2\pi \]2-qadam
Mana bu egri chiziq uchun integrali.
\[\begin{align*}\int\limits_{C}{{\left( {x - y} \right)\,dx - y{x^2}\,dy}} & = \int\limits_{C}{{\left( {x - y} \right)\,dx}} - \int\limits_{C}{{y{x^2}\,dy}}\\ & = \int_{{\frac{1}{2}\pi }}^{{2\pi }}{{\left[ {6\cos \left( t \right) - \left( { - 6\sin \left( t \right)} \right)} \right]\left( { - 6\sin \left( t \right)} \right)\,dt}}\\ & \hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\,\,\,\,\,\, - \int_{{\frac{1}{2}\pi }}^{{2\pi }}{{\left[ {\left( {6\sin \left( t \right)} \right){{\left( {6\cos \left( t \right)} \right)}^2}} \right]\left( { - 6\cos \left( t \right)} \right)\,dt}}\\ & = \int_{{\frac{1}{2}\pi }}^{{2\pi }}{{ - 36\cos \left( t \right)\sin \left( t \right) - 36{{\sin }^2}\left( t \right) - 1296\sin \left( t \right){{\cos }^3}\left( t \right)\,dt}}\\ & = \int_{{\frac{1}{2}\pi }}^{{2\pi }}{{ - 18\sin \left( {2t} \right) - 18\left( {1 - \cos \left( {2t} \right)} \right) - 1296\sin \left( t \right){{\cos }^3}\left( t \right)\,dt}}\\ & = \left. {\left( {9\cos \left( {2t} \right) - 18t + 9\sin \left( {2t} \right) + 324{{\cos }^4}\left( t \right)} \right)} \right|_{\frac{1}{2}\pi }^{2\pi }\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{342 - 27\pi = 257.1770}}\end{align*}\]Chiziqli integralni "standart" integralga o'tkazishda differensiallar bilan to'g'ri ishlashni unutmang.
b \(C\) - \(\left( {0, - 6} \right)\) dan \(\left( {6,0} \right)\) gacha bo'lgan chiziq segmenti.
Yechim
Keling, muammoning ushbu qismi uchun egri chiziqning qisqacha chizmasidan boshlaylik.
Shunday qilib, bizda bu qismda \(\left( {0, - 6} \right)\) dan \(\left( {6,0} \right)\) gacha bo'lgan boshqa egri chiziq mavjud. Ushbu egri chiziq birinchi qismdagi egri chiziq bilan bir xil boshlang'ich va tugash nuqtasiga ega bo'lishiga qaramay, chiziqli integrali bir xil qiymatga ega bo'lishini kutish uchun hech qanday sabab yo'q. Shuning uchun biz ishni ko'rib chiqishimiz va chiziqli integraldan nimani olishimizni ko'rishimiz kerak.
Biz egri chiziqni parametrlashimiz kerak, shuning uchun bu haqda g'amxo'rlik qilaylik.
\[C:\,\,\vec r\left( t \right) = \left( {1 - t} \right)\left\langle {0, - 6} \right\rangle + t\left\langle {6,0} \right\rangle = \left\langle {6t, - 6 + 6t} \right\rangle \hspace{0.25in} 0 \le t \le 1\]E'tibor bering, biz bu egri chiziqning tenglamasini topishimiz mumkin edi, lekin ikkita nuqta orasidagi chiziq segmentining vektor shaklini ishlatish juda oson tuyuldi.
2-qadam
Endi biz faqat chiziq integralini hisoblashimiz kerak.
\[\begin{align*}\int\limits_{C}{{\left( {x - y} \right)\,dx - y{x^2}\,dy}} & = \int\limits_{C}{{\left( {x - y} \right)\,dx}} - \int\limits_{C}{{y{x^2}\,dy}}\\ & = \int_{0}^{1}{{\left[ {6t - \left( { - 6 + 6t} \right)} \right]\left( 6 \right)\,dt}} - \int_{0}^{1}{{\left[ {\left( { - 6 + 6t} \right){{\left( {6t} \right)}^2}} \right]\left( 6 \right)\,dt}}\\ & = \int_{0}^{1}{{36 + 1296{t^2} - 1296{t^3}\,dt}}\\ & = \left. {\left( {36t + 432{t^3} - 324{t^4}} \right)} \right|_0^1 = \require{bbox} \bbox[2pt,border:1px solid black]{{144}}\end{align*}\]Shunday qilib, ushbu qismning boshida ta'kidlanganidek, egri chiziqning bir xil boshlang'ich va tugash nuqtalariga qaramay, chiziqli integralning qiymati birinchi qismdagi chiziqli integralning qiymati bilan bir xil emas edi. E'tibor bering, bir xil boshlang'ich va tugash nuqtalari bo'lgan ikkita chiziqli integral bir xil qiymatga ega bo'lishi mumkin, ammo biz bunday bo'lishini kuta olmaymiz va shuning uchun ishni bajarishimiz kerak.