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15.6-bo'lim : Silindrsimon koordinatalarda uch karrali integrallar

  1. \( \displaystyle \iiint\limits_{E}{{8z\,dV}}\) ni baholang, bu erda \(E\) 1-oktantda \(z = 2{x^2} + 2{y^2} - 4\) va \(z = 5 - {x^2} - {y^2}\) bilan chegaralangan mintaqadir.
  2. \( \displaystyle \iiint\limits_{E}{{6xy\,dV}}\) ni baholang, bunda \(E\) \(z = 2x - 10\) ustidagi, \(z = 2\) ostida va \({x^2} + {y^2} = 4\) silindrining ichida joylashgan.
  3. \( \displaystyle \iiint\limits_{E}{{9y{z^3}\,dV}}\) ni baholang, bu erda \(E\) \({y^2} + {z^2} = 1\) silindridagi \(x = - \sqrt {9{y^2} + 9{z^2}} \) va \(x = \sqrt {{y^2} + {z^2}} \) orasidagi hududdir.
  4. \( \displaystyle \iiint\limits_{E}{{x + 2\,dV}}\) ni baholang, bunda \(E\) \(x = 18 - 4{y^2} - 4{z^2}\) va \(x = 2\) \(z \ge 0\) bilan chegaralangan hududdir.
  5. \( \displaystyle \iiint\limits_{E}{{x + 2\,dV}}\) ni baholang, bu erda \(E\) \({x^2} + {z^2} = 16\) silindridagi ikkita \(2x + y + z = 6\) va \(6x + 3y + 3z = 12\) tekisliklari orasidagi hududdir.
  6. \( \displaystyle \iiint\limits_{E}{{{x^2}\,dV}}\) ni baholang, bu erda \(E\) \(y = {x^2} + {z^2} - 4\) va \(y = 8 - 5{x^2} - 5{z^2}\) \(x \le 0\) bilan chegaralangan hududdir.
  7. 1-oktantda \(z = \sqrt {{x^2} + {y^2}} \) va \(z = {x^2} + {y^2}\) bilan chegaralangan mintaqa hajmini aniqlash uchun uch karrali integraldan foydalaning.
  8. 1-oktantda \(y = \sqrt {9{x^2} + 9{z^2}} \) va \(y = - 3{x^2} - 3{z^2}\) bilan chegaralangan hudud hajmini aniqlash uchun uch karrali integraldan foydalaning.
  9. \(x = z + 3\) orqasidagi, \(x = - z - 6\) oldidagi va \({y^2} + {z^2} = 4\) silindrining ichidagi hudud hajmini aniqlash uchun uch karrali integraldan foydalaning.
  10. Avval silindrsimon koordinatalarda integralga aylantirib, quyidagi integralni baholang. \[\int_{{ - 4}}^{4}{{\int_{0}^{{\sqrt {16 - {y^{\,2}}} }}{{\int_{0}^{{6 + x}}{{\,\,\,6y{x^2}\,\,dz}}\,dx}}\,dy}}\]
  11. Avval silindrsimon koordinatalarda integralga aylantirib, quyidagi integralni baholang. \[\int_{0}^{3}{{\int_{{ - \sqrt {9 - {x^{\,2}}} }}^{{\sqrt {9 - {x^{\,2}}} }}{{\int_{{ - \sqrt {2{x^{\,2}} + 2{y^{\,2}}} }}^{{6 + {x^{\,2}} + {y^{\,2}}}}{{\,\,\,15z\,\,dz}}\,dy}}\,dx}}\]
  12. Balandligi \(h\) va \(a\) radiusli silindr hajmini olish uchun silindrsimon koordinatalarda uch karrali integraldan foydalaning.