LineIntegralsPtII - Vazifa topshiriqlari | MathModel.uz

16.3-bo'lim : Chiziqli integrallar - II qism

1 – 7 masalalar uchun berilgan chiziqli integralni baholang. Muammo bayonotida ko'rsatilganidek, $C$ yo'nalishiga amal qiling.

$ \displaystyle \int\limits_{C}{{xy\,dx + \left( {x - y} \right)\,dy}}$-ni baholang, bu erda $C$ $\left( {0, - 3} \right)$dan $\left( { - 4,1} \right)$gacha bo'lgan chiziq segmentidir.
$ \displaystyle \int\limits_{C}{{{{\bf{e}}^{3x}}\,dx}}$ni baholang, bunda $C$ $x = \sin \left( {4y} \right)$ ning $\displaystyle y = \frac{\pi }{8}$ dan $y = \pi $ gacha bo‘lgan qismidir.
$ \displaystyle \int\limits_{C}{{x\,dy - \left( {{x^2} + y} \right)\,dx}}$ ni baholang, bunda $C$ aylananing 2 da 3 radiusning boshiga markazlashgan qismidir.^nd soat yo'nalishi bo'yicha aylanish bilan kvadrant.
$ \displaystyle \int\limits_{C}{{dx - 3{y^3}\,dy}}$ni baholang, bu erda $C$ $\vec r\left( t \right) = 4\sin \left( {\pi t} \right)\,\,\vec i + {\left( {t - 1} \right)^2}\vec j$ tomonidan $0 \le t \le 1$ bilan berilgan.
$ \displaystyle \int\limits_{C}{{4{y^2}\,dx + 3x\,dy + 2z\,dz}}$ni baholang, bu erda $C$ $\left( {4, - 1,2} \right)$ dan $\left( {1,7, - 1} \right)$ gacha bo'lgan chiziq segmentidir.
$ \displaystyle \int\limits_{C}{{\left( {yz + x} \right)dx + yz\,dy\, - \left( {y + z} \right)\,dz}}$ni baholang, bu erda $C$ $\vec r\left( t \right) = 3t\,\vec i + 4\sin \left( t \right)\vec j + 4\cos \left( t \right)\,\vec k$ tomonidan $0 \le t \le \pi $ bilan berilgan.
$ \displaystyle \int\limits_{C}{{7xy\,dy}}$ ni baholang, bunda $C$ $y = \sqrt {{x^2} + 5} $ ning $x = - 1$ dan $x = 2$ gacha bo‘lgan qismi va keyin $\left( {2,3} \right)$ dan $\left( {4, - 1} \right)$ gacha bo‘lgan chiziq segmenti. Yo'nalish uchun quyidagi eskizga qarang.
$This curve starts with the portion of $y=\sqrt[x^{2}+5]$ starting at $\left( -1, \sqrt[6] \right)$ and ending at (2,3) and followed by a line starting at (2,3) and ending at (4,-1).$
$ \displaystyle \int\limits_{C}{{\left( {{y^2} - x} \right)\,dx - 4y\,dy}}$ ni baholang, bunda $C$ $y = {x^2}$ ning $x = - 2$ dan $x = 2$ gacha bo‘lgan qismi, undan keyin $\left( {2,4} \right)$ dan $\left( {0,6} \right)$ gacha bo‘lgan chiziq segmenti, o‘z navbatida $\left( {0,6} \right)$ dan $\left( { - 2,4} \right)$ gacha bo‘lgan chiziq segmenti. Yo'nalish uchun quyidagi eskizga qarang.
$This curve starts with the portion of $y=x^{2}$ starting at (-2,4) and ending at (2,4). This is followed by a line starting at (2,4) and ending at (0,6). The final portion of the curve is a line starting at (0,6) and ending at (-2,4).$
Quyidagi egri chiziqlarning har biri uchun $ \displaystyle \int\limits_{C}{{\left( {{x^2} - 2} \right)\,dx + 7x{y^2}\,dy}}$ ni baholang.
1. $C$ - $x = - {y^2}$ ning $y = - 1$ dan $y = 1$ gacha bo'lgan qismi.
2. $C$ - $\left( { - 1, - 1} \right)$ dan $\left( {1,1} \right)$ gacha bo'lgan chiziq segmenti.
Quyidagi egri chiziqlarning har biri uchun $ \displaystyle \int\limits_{C}{{{x^3} + 9y\,dy}}$ ni baholang.
1. $C$ - $y = 1 - {x^2}$ ning $x = - 1$ dan $x = 1$ gacha bo'lgan qismi.
2. $C$ - $\left( { - 1,0} \right)$ dan $\left( {0, - 1} \right)$ gacha bo'lgan chiziq segmenti va undan keyin $\left( {0, - 1} \right)$ dan $\left( {1,0} \right)$ gacha bo'lgan chiziq segmenti.
Quyidagi egri chiziqlarning har biri uchun $ \displaystyle \int\limits_{C}{{x{y^3}\,dx - 4x\,dy}}$ ni baholang.
1. $C$ aylananing 1 da radius 7 ning boshida joylashgan qismidir.^st soat miliga teskari aylanish bilan kvadrant.
2. $C$ - aylananing 1 da radius 7 ning boshida joylashgan qismi.^st soat yo'nalishi bo'yicha aylanish bilan kvadrant.