LineIntegralsPtII - Amaliy mashqlar | MathModel.uz

16.3-bo'lim: Chiziqli integrallar - II qism

1-5-masalalar uchun berilgan chiziqli integralni baholang. Muammo bayonotida ko'rsatilganidek, $C$ yo'nalishiga amal qiling.

$ \displaystyle \int\limits_{C}{{\sqrt {1 + y} \,dy}}$ ni baholang, bu erda $C$ $y = {{\bf{e}}^{2x}}$ ning $x = 0$ dan $x = 2$ gacha bo'lgan qismidir. Yechim
$ \displaystyle \int\limits_{C}{{2y\,dx + \left( {1 - x} \right)\,dy}}$ni baholang, bunda $C$ $y = 1 - {x^3}$ ning $x = - 1$ dan $x = 2$ gacha bo‘lgan qismidir. Yechim
$ \displaystyle \int\limits_{C}{{{x^2}\,dy - yz\,dz}}$ni baholang, bu erda $C$ $\left( {4, - 1,2} \right)$ dan $\left( {1,7, - 1} \right)$ gacha bo'lgan chiziq segmentidir. Yechim
$ \displaystyle \int\limits_{C}{{1 + {x^3}\,dx}}$ ni baholang, bunda $C$ radius 2 aylanasining o‘ng yarmi soat miliga teskari aylanib, undan keyin $\left( {0,2} \right)$ dan $\left( { - 3, - 4} \right)$ gacha bo‘lgan chiziq segmenti. Yo'nalish uchun quyidagi eskizga qarang.
$The curve $C$. The arrows show that it starts at $\left( {0, - 2} \right)$ and travels counter clockwise around the right half of the circle of radius 2 centered at the origin, through $\left( {2,0} \right)$, to $\left( {0,2} \right)$. From there it runs down and to the left along a straight line to $\left( { - 3, - 4} \right)$.$
Yechim
$ \displaystyle \int\limits_{C}{{2{x^2}\,dy - xy\,dx}}$ ni baholang, bunda $C$ $\left( {1, - 5} \right)$ dan $\left( { - 2, - 3} \right)$ gacha bo‘lgan chiziq segmenti, so‘ngra $y = 1 - {x^2}$ ning $x = - 2$ dan $x = 2$ gacha bo‘lgan qismi, o‘z navbatida $\left( {2, - 3} \right)$ dan $y = {{\bf{e}}^{2x}}$ ga bo‘lgan chiziq segmenti. Yo'nalish uchun quyidagi eskizga qarang.
$The curve $C$. The arrows show that it starts at $\left( {1, - 5} \right)$ and runs up and to the left along a straight line to $\left( { - 2, - 3} \right)$. From there it follows the parabola $y = 1 - {x^2}$, up over the peak at $\left( {0,1} \right)$ and back down to $\left( {2, - 3} \right)$, and then runs right along a horizontal line to $\left( {4, - 3} \right)$.$
Yechim
Quyidagi egri chiziqlarning har biri uchun $ \displaystyle \int\limits_{C}{{\left( {x - y} \right)\,dx - y{x^2}\,dy}}$ ni baholang.
1. $C$ - radiusi 6 bo'lgan aylananing 1 dagi qismi^st, 2^nd va 3^rd soat yo'nalishi bo'yicha aylanish bilan kvadrant.
2. $C$ - $\left( {0, - 6} \right)$ dan $\left( {6,0} \right)$ gacha bo'lgan chiziq segmenti.
Yechim
Quyidagi egri chiziqlarning har biri uchun $ \displaystyle \int\limits_{C}{{{x^3}\,dy - \left( {y + 1} \right)\,dx}}$ ni baholang.
1. $C$ - $\left( {1,7} \right)$ dan $\left( { - 2,4} \right)$ gacha bo'lgan chiziq segmenti.
2. $C$ - $\left( { - 2,4} \right)$ dan $\left( {1,7} \right)$ gacha bo'lgan chiziq segmenti.
Yechim