Asosiy tarkibga o'tish

16.2-bo'lim : Chiziqli integrallar - I qism

1 – 10 masalalar uchun berilgan chiziqli integralni baholang. Muammo bayonotida ko'rsatilganidek, \(C\) yo'nalishiga amal qiling.

  1. \( \displaystyle \int\limits_{C}{{3y\,ds}}\) ni baholang, bunda \(C\) \(y = - 1\) va \(y = 2\) dan \(x = 9 - {y^2}\) qismidir.
  2. \( \displaystyle \int\limits_{C}{{\sqrt x + 2xy\,ds}}\) ni baholang, bu erda \(C\) \(\left( {7,3} \right)\) dan \(\left( {0,6} \right)\) gacha bo'lgan chiziq segmentidir.
  3. \( \displaystyle \int\limits_{C}{{{y^2} - 10xy\,ds}}\) ni baholang, bu erda \(C\) soat miliga teskari aylanish bilan 6 radiusning boshida joylashgan aylananing chap yarmidir.
  4. \( \displaystyle \int\limits_{C}{{{x^2} - 2y\,ds}}\)ni baholang, bu erda \(C\) \(\vec r\left( t \right) = \left\langle {4{t^4},{t^4}} \right\rangle \) tomonidan \( - 1 \le t \le 0\) uchun berilgan.
  5. \( \displaystyle \int\limits_{C}{{{z^3} - 4x + 2y\,ds}}\)ni baholang, bu erda \(C\) \(\left( {2,4, - 1} \right)\) dan \(\left( {1, - 1,0} \right)\) gacha bo'lgan chiziq segmentidir.
  6. 6. \( \displaystyle \int\limits_{C}{{x + 12xz\,ds}}\) ni baholang, bu erda \(C\) \(\displaystyle \vec r\left( t \right) = \left\langle {t,\frac{1}{2}{t^2},\frac{1}{4}{t^4}} \right\rangle \) tomonidan \( - 2 \le t \le 1\) uchun berilgan.
  7. \( \displaystyle \int\limits_{C}{{{z^3}\left( {x + 7} \right) - 2y\,ds}}\) ni baholang, bunda \(C\) - \(x = - 3\) da \(x\) o'qi markazida joylashgan 1 radiusning boshida joylashgan aylana. Yo'nalish uchun quyidagi eskizlarga qarang.
    This is a sketch with the standard 3D coordinate system. The positive z-axis is straight up, the positive x-axis moves off to the left and slightly downward and positive y-axis moves off the right and slightly downward. The circle centered on the x-axis at x=-3. If you are standing on the positive x-axis and looking towards the circle there is a clockwise orientation to the circle. This is a 2D sketch of the circle. In this case think of the x-axis as coming directly out of the page and is not shown. The positive z-axis is pointed upwards from the origin and the positive y-axis is pointed right out of the origin. The circle has a clockwise orientation.
  8. \( \displaystyle \int\limits_{C}{{6x\,ds}}\) ni baholang, bunda \(C\) \(y = 3 + {x^2}\) ning \(x = - 2\) dan \(x = 0\) gacha bo‘lgan qismi, so‘ngra \(y = 3 - {x^2}\) shaklining \(x = 0\) dan \(x = 2\) ga bo‘lgan qismi, o‘z navbatida MABLOCK37X dan MABLOCK37X ga bo‘lgan segment. Yo'nalish uchun quyidagi eskizga qarang.
    The full curve starts with the graph of $y=3+x^{2}$ starting at (-2,7) and ending at (0,3). The next portion is the graph of $y=3-x^{2}$ starting at (0,3) and ending at (2,-1). The final portion is a line starting at (2,-1) and ending at (-1,-2).
  9. \( \displaystyle \int\limits_{C}{{2 - xy\,ds}}\) ni baholang, bunda \(C\) 1 radiusning boshiga markazlashgan doiraning yuqori yarmi soat yo‘nalishi bo‘yicha aylanib, undan keyin \(\left( {1,0} \right)\) dan \(\left( {3,0} \right)\) gacha bo‘lgan chiziq segmenti, o‘z navbatida soat yo‘nalishi bo‘yicha 3 radiusning boshiga markazlashgan doiraning pastki yarmidan keyin keladi. Yo'nalish uchun quyidagi eskizga qarang.
    The full curve starts with upper part of the circle centered at radius 1 centered at the origin with clockwise orientation. This is followed by a line from (1,0) to (3,0). Finally, there is the lower half of a circle centered at radius 3 centered at the origin with counter clockwise orientation.
  10. \( \displaystyle \int\limits_{C}{{3xy + {{\left( {x - 1} \right)}^2}\,ds}}\) ni baholang, bunda \(C\) uchburchak \(\left( {0,3} \right)\), \(\left( {6,0} \right)\) va \(\left( {0,0} \right)\) soat yo'nalishi bo'yicha aylantiriladi.
  11. Quyidagi egri chiziqlarning har biri uchun \( \displaystyle \int\limits_{C}{{{x^5}\,ds}}\) ni baholang.
    1. \(C\) - \(\left( { - 1,3} \right)\) dan \(\left( {0,0} \right)\) gacha bo'lgan chiziq segmenti va undan keyin \(\left( {0,0} \right)\) dan \(\left( {0,4} \right)\) gacha bo'lgan chiziq segmenti.
    2. \(C\) - \(y = 4 - {x^4}\) ning \(x = - 1\) dan \(x = 0\) gacha bo'lgan qismi.
  12. Quyidagi egri chiziqlarning har biri uchun \( \displaystyle \int\limits_{C}{{3x - 6y\,ds}}\) ni baholang.
    1. \(C\) - \(\left( {6,0} \right)\) dan \(\left( {0,3} \right)\) gacha bo'lgan chiziq segmenti va undan keyin \(\left( {0,3} \right)\) dan \(\left( {6,6} \right)\) gacha bo'lgan chiziq segmenti.
    2. \(C\) - \(\left( {6,0} \right)\) dan \(\left( {6,6} \right)\) gacha bo'lgan chiziq segmenti.
  13. Quyidagi egri chiziqlarning har biri uchun \( \displaystyle \int\limits_{C}{{{y^2} - 3z + 2\,ds}}\) ni baholang.
    1. \(C\) - \(\left( {1,0,4} \right)\) dan \(\left( {2, - 1,1} \right)\) gacha bo'lgan chiziq segmenti.
    2. \(C\) - \(\left( {2, - 1,1} \right)\) dan \(\left( {1,0,4} \right)\) gacha bo'lgan chiziq segmenti.