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16.2-bo'lim: Chiziqli integrallar - I qism

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

  1. \( \displaystyle \int\limits_{C}{{3{x^2} - 2y\,ds}}\)-ni baholang, bu erda \(C\) \(\left( {3,6} \right)\)dan \(\left( {1, - 1} \right)\)gacha bo'lgan chiziq segmentidir. Yechim
  2. \( \displaystyle \int\limits_{C}{{2y{x^2} - 4x\,ds}}\) ni baholang, bu erda \(C\) soat yo'nalishi bo'yicha aylanish bilan 3 radiusning boshida joylashgan doiraning pastki yarmidir. Yechim
  3. \( \displaystyle \int\limits_{C}{{6x\,ds}}\) ni baholang, bunda \(C\) \(y = {x^2}\) ning \(x = - 1\) dan \(x = 2\) gacha bo‘lgan qismidir. \(C\) yo'nalishi \(x\) ni oshirish yo'nalishida. Yechim
  4. \( \displaystyle \int\limits_{C}{{xy - 4z\,ds}}\)ni baholang, bu erda \(C\) \(\left( {1,1,0} \right)\) dan \(\left( {2,3, - 2} \right)\) gacha bo'lgan chiziq segmentidir. Yechim
  5. \( \displaystyle \int\limits_{C}{{{x^2}{y^2}\,ds}}\) ni baholang, bu erda \(C\) - \(y = 4\) da \(y\) o'qida markazlashtirilgan 2 radiusning boshida joylashgan aylana. Orientatsiya uchun quyidagi eskizlarga qarang. E'tibor bering, 2D doiradagi "g'alati" o'q yo'nalishi ataylab 3D o'qi yo'nalishiga mos keladi.
    A three dimensional sketch of the curve \(C\). It is a circle of radius 2 that lies in a plane perpendicular to the y-axis and is centered on the y-axis at \(y = 4\). Arrows on the circle show the direction of travel to be from the positive z-axis moving towards the positive x-axis. A two dimensional sketch of the same circle of radius 2 as seen looking down the y-axis, so that the circle appears centered at the origin. Note that the positive x-axis has been drawn pointing to the left so that the orientation matches the 3D sketch. The arrows show that the circle is traveled counter clockwise as it is drawn here.
    Yechim
  6. \( \displaystyle \int\limits_{C}{{16{y^5}\,ds}}\)ni baholang, bunda \(C\) \(x = {y^4}\) ning \(y = 0\) dan \(y = 1\) gacha bo‘lgan qismi, keyin \(\left( {1,1} \right)\) dan \(\left( {1, - 2} \right)\) gacha bo‘lgan chiziq segmenti, o‘z navbatida \(\left( {1, - 2} \right)\) dan \(\left( {3,6} \right)\) ga bo‘lgan chiziq segmenti. Yo'nalish uchun quyidagi eskizga qarang.
    The curve \(C\). The arrows show that it starts at the origin and follows the curve \(x = {y^4}\) up to \(\left( {1,1} \right)\), then drops straight down the line \(x = 1\) to \(\left( {1, - 2} \right)\) and finally runs up and to the right along a straight line to \(\left( {2,0} \right)\).
    Yechim
  7. \( \displaystyle \int\limits_{C}{{4y - x\,ds}}\) ni baholang, bunda \(C\) 3 radiusning boshida markazlashtirilgan aylananing yuqori qismi \(\displaystyle\left( {\frac{3}{{\sqrt 2 }},\frac{3}{{\sqrt 2 }}} \right)\) dan \(\displaystyle\left( { - \frac{3}{{\sqrt 2 }}, - \frac{3}{{\sqrt 2 }}} \right)\) gacha soat miliga teskari aylanishda, keyin esa \(\displaystyle\left( { - \frac{3}{{\sqrt 2 }}, - \frac{3}{{\sqrt 2 }}} \right)\) dan \(\displaystyle\left( {4, - \frac{3}{{\sqrt 2 }}} \right)\) gacha bo‘lgan chiziq segmenti bo‘lib, o‘z navbatida MABLOCK 4 dan MABLX 5 ga bo‘lgan qatorga o‘tadi. \(\left( {4,4} \right)\). Yo'nalish uchun quyidagi eskizga qarang.
    The curve \(C\). The arrows show that it starts at \(\left( {\frac{3}{{\sqrt 2 }},\frac{3}{{\sqrt 2 }}} \right)\) and travels counter clockwise around the circle of radius 3 centered at the origin, up over the top and around the left side, to \(\left( { - \frac{3}{{\sqrt 2 }}, - \frac{3}{{\sqrt 2 }}} \right)\). From there it runs right along a horizontal line to \(\left( {4, - \frac{3}{{\sqrt 2 }}} \right)\) and then straight up to \(\left( {4,4} \right)\).
    Yechim
  8. Quyidagi egri chiziqlarning har biri uchun \( \displaystyle \int\limits_{C}{{{y^3} - {x^2}\,ds}}\) ni baholang.
    1. \(C\) - \(\left( {3,6} \right)\) dan \(\left( {0,0} \right)\) gacha bo'lgan chiziq segmenti va undan keyin \(\left( {0,0} \right)\) dan \(\left( {3, - 6} \right)\) gacha bo'lgan chiziq segmenti.
    2. \(C\) - \(\left( {3,6} \right)\) dan \(\left( {3, - 6} \right)\) gacha bo'lgan chiziq segmenti.
    Yechim
  9. Quyidagi egri chiziqlarning har biri uchun \( \displaystyle \int\limits_{C}{{4{x^2}\,ds}}\) ni baholang.
    1. \(C\) - aylananing 1 da 2 radiusning boshida joylashgan qismi.st soat yo'nalishi bo'yicha aylanadigan kvadrant.
    2. \(C\) - \(\left( {0,2} \right)\) dan \(\left( {2,0} \right)\) gacha bo'lgan chiziq segmenti.
    Yechim
  10. Quyidagi egri chiziqlarning har biri uchun \( \displaystyle \int\limits_{C}{{2{x^3}\,ds}}\) ni baholang.
    1. \(C\) - \(y = {x^3}\) dan \(x = - 1\) dan \(x = 2\) gacha bo'lgan qism.
    2. \(C\) - \(y = {x^3}\) qismi \(x = 2\) dan \(x = - 1\) gacha.
    Yechim