15.6-bo'lim: Silindrsimon koordinatalarda uch karrali integrallar

1. \( \displaystyle \iiint\limits_{E}{{4xy\,dV}}\) ni baholang, bu erda \(E\) \(z = 2{x^2} + 2{y^2} - 7\) va \(z = 1\) bilan chegaralangan mintaqadir. Yechim
2. \( \displaystyle \iiint\limits_{E}{{{{\bf{e}}^{ - {x^{\,2}} - {z^{\,2}}}}\,dV}}\) ni baholang, bunda \(E\) \(1 \le y \le 5\) va \(z \le 0\) bilan ikkita \({x^2} + {z^2} = 4\) va \({x^2} + {z^2} = 9\) silindrlari orasidagi hududdir. Yechim
3. \( \displaystyle \iiint\limits_{E}{{z\,dV}}\) ni baholang, bu erda \(E\) \(x + y + z = 2\) va \(x = 0\) ikkita tekislik orasidagi va \({y^2} + {z^2} = 1\) silindrining ichidagi mintaqadir. Yechim
4. \(x \le 0\) bilan \({x^2} + {y^2} = 3\) silindr ichidagi \(z = 6 - x\) ostida, \(z = - \sqrt {4{x^2} + 4{y^2}} \) ustidagi mintaqa hajmini aniqlash uchun uch karrali integraldan foydalaning. Yechim
5. Avval silindrsimon koordinatalarda integralga aylantirib, quyidagi integralni baholang. \[\int_{0}^{{\sqrt 5 }}{{\int_{{ - \sqrt {5 - {x^{\,2}}} }}^{0}{{\int_{{{x^{\,2}} + {y^{\,2}} - 11}}^{{9 - 3{x^{\,2}} - 3{y^{\,2}}}}{{\,\,\,2x - 3y\,\,\,dz}}\,dy}}\,dx}}\] Yechim