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15.4-bo'lim : Qutb koordinatalarida ikki karrali integrallar

  1. \( \displaystyle \iint\limits_{D}{{3x{y^2} - 2\,dA}}\)-ni baholang, bu erda \(D\) - boshda markazlashtirilgan birlik doirasi.
  2. \( \displaystyle \iint\limits_{D}{{4x - 2y\,dA}}\) ni baholang, bu erda \(D\) \({x^2} + {y^2} = 4\) va \({x^2} + {y^2} = 25\) o'rtasidagi mintaqaning yuqori yarmidir.
  3. \( \displaystyle \iint\limits_{D}{{6xy + 4{x^2}\,dA}}\) ni baholang, bu erda \(D\) 2-kvadrantdagi \({x^2} + {y^2} = 9\) qismidir.
  4. \( \displaystyle \iint\limits_{D}{{\sin \left( {3{x^2} + 3{y^2}} \right)\,dA}}\) ni baholang, bu erda \(D\) \({x^2} + {y^2} = 1\) va \({x^2} + {y^2} = 7\) o'rtasidagi mintaqadir.
  5. \( \displaystyle \iint\limits_{D}{{{{\bf{e}}^{1 - {x^{\,2}} - {y^{\,2}}}}\,dA}}\) ni baholang, bu erda \(D\) \({x^2} + {y^2} = 16\) va \({x^2} + {y^2} = 36\) o'rtasidagi 4-kvadrantdagi mintaqadir.
  6. \(r = 6 - 4\cos \theta \) ichidagi mintaqa maydonini aniqlash uchun ikki karrali integraldan foydalaning.
  7. \(r = 4\) ichidagi va \(r = 8 + 6\sin \theta \) tashqarisidagi mintaqa maydonini aniqlash uchun ikki karrali integraldan foydalaning.
  8. Avval qutb koordinatalarida integralga aylantirib, quyidagi integralni baholang. \[\int_{{ - 2}}^{0}{{\int_{{ - \sqrt {4 - {y^{\,2}}} }}^{{\sqrt {4 - {y^{\,2}}} }}{{\,\,\,{x^2}\,dx}}\,dy}}\]
  9. Avval qutb koordinatalarida integralga aylantirib, quyidagi integralni baholang. \[\int_{{ - 1}}^{1}{{\int_{0}^{{\sqrt {1 - {x^{\,2}}} }}{{\,\,\,\sqrt {{x^2} + {y^2}} \,dy}}\,dx}}\]
  10. \(z = 9 - 4{x^2} - 4{y^2}\) ostidagi va \(xy\) tekisligidan yuqori bo'lgan qattiq jismning hajmini aniqlash uchun ikki karrali integraldan foydalaning.
  11. \(z = 12 - 3{x^2} - 3{y^2}\) va \(z = {x^2} + {y^2} - 8\) bilan chegaralangan qattiq jismning hajmini aniqlash uchun ikki karrali integraldan foydalaning.
  12. \({x^2} + {y^2} = 9\) tsilindri va \({x^2} + {y^2} + {z^2} = 16\) shari ichidagi qattiq jismning hajmini aniqlash uchun ikki karrali integraldan foydalaning.
  13. \(a\) radiusi bo'lgan doiraning maydonini olish uchun ikki karrali integraldan foydalaning.
  14. \(\alpha \le \theta \le \beta \) bilan a va b radiusli doiralar orasidagi mintaqa maydonini olish uchun ikki karrali integraldan foydalaning. Mintaqaning eskizi uchun quyidagi rasmga qarang.
    The region is the portion of a ring in the first quadrant. The inner radius of the ring is given as “a” and the outer radius of the ring is given as “b”. The ring starts at a line that makes an angle of $\alpah$ with the positive x-axis and ends at a line that makes an angle of $\beta$ with the positive x-axis where $\alpha < $\beta$.