GreensTheorem - 2-masala yechimi | MathModel.uz

16.7-bo'lim : Grin teoremasi

2. $ \displaystyle \int\limits_{C}{{\left( {6y - 9x} \right)dy - \left( {yx - {x^3}} \right)\,dx}}$ ni baholash uchun Grin teoremasidan foydalaning, bu erda $C$ quyida ko'rsatilgan.

$The curve $C$. It is a closed four sided figure with corners at $\left( { - 1,4} \right)$, $\left( { - 1, - 1} \right)$, $\left( {1, - 1} \right)$ and $\left( {1,2} \right)$. The arrows show that it is traveled in the counter clockwise direction, going down the left side, right along the bottom, up the right side and then back up along the slanted top side to $\left( { - 1,4} \right)$.$

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,

\[P = - \left( {yx - {x^3}} \right) = {x^3} - yx\hspace{0.25in}\hspace{0.25in}Q = 6y - 9x\]

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. Shoshilib ketish ham oson va faqat $P$ integralning birinchi hadi, $Q$ ikkinchisi deb taxmin qilish mumkin. Bu erda bunday emasligi aniq, shuning uchun ehtiyot bo'ling!

2-qadam

Green teoremasidan foydalanib, chiziqli integralga aylanadi,

\[\int\limits_{C}{{\left( {6y - 9x} \right)dy - \left( {yx - {x^3}} \right)\,dx}} = \iint\limits_{D}{{ - 9 - \left( { - x} \right)\,dA}} = \iint\limits_{D}{{x - 9\,dA}}\]

$D$ - egri chiziq bilan o'ralgan hudud.

3-qadam

Mintaqaning yuqori qismidagi chiziq tenglamasi $y = 3 - x$ tomonidan berilganligini tekshirish uchun buni sizga qoldiramiz. Ushbu tenglamaga ega bo'lganimizdan so'ng, mintaqa uchun chegaralarni olish juda oson. Ular,

\[\begin{array}{c} - 1 \le x \le 1\\ - 1 \le y \le 3 - x\end{array}\]

4-qadam

Endi biz ikki karrali integralni baholashimiz kerak. Bu erda baholash ishi.

\[\begin{align*}\int\limits_{C}{{\left( {6y - 9x} \right)dy - \left( {yx - {x^3}} \right)\,dx}} & = \iint\limits_{D}{{x - 9\,dA}}\\ & = \int_{{ - 1}}^{1}{{\int_{{ - 1}}^{{3 - x}}{{x - 9\,dy}}\,dx}}\\ & = \int_{{ - 1}}^{1}{{\left. {\left( {x - 9} \right)y} \right|_{ - 1}^{3 - x}\,dx}}\\ & = \int_{{ - 1}}^{1}{{\left( {x - 9} \right)\left( {4 - x} \right)\,dx}}\\ & = \int_{{ - 1}}^{1}{{ - {x^2} + 13x - 36\,dx}}\\ & = \left. {\left[ { - \frac{1}{3}{x^3} + \frac{{13}}{2}{x^2} - 36x} \right]} \right|_{ - 1}^1 = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{{218}}{3}}}\end{align*}\]