15.7-bo'lim : Sferik koordinatalarda uch karrali integrallar

1. \( \displaystyle \iiint\limits_{E}{{4{y^2}\,dV}}\) ni baholang, bu erda \(E\) \({x^2} + {y^2} + {z^2} = 9\) sharidir.
2. \( \displaystyle \iiint\limits_{E}{{3x - 2y\,dV}}\) ni baholang, bunda \(E\) \(z \le 0\) bilan \({x^2} + {y^2} + {z^2} = 1\) va \({x^2} + {y^2} + {z^2} = 4\) sharlari orasidagi mintaqadir.
3. \( \displaystyle \iiint\limits_{E}{{2yz\,dV}}\) ni baholang, bu erda \(E\) \({x^2} + {y^2} + {z^2} = 16\) va \(z = \sqrt {3{x^2} + 3{y^2}} \) ichidagi mintaqa bo'lib, 1-oktantda joylashgan.
4. \( \displaystyle \iiint\limits_{E}{{{z^2}\,dV}}\) ni baholang, bu erda \(E\) \({x^2} + {y^2} + {z^2} = 4\) va \({x^2} + {y^2} + {z^2} = 25\) sharlari orasidagi va \(\displaystyle z = - \sqrt {\frac{1}{3}{x^2} + \frac{1}{3}{y^2}} \) ichidagi mintaqadir.
5. \( \displaystyle \iiint\limits_{E}{{5{y^2}\,dV}}\) ni baholang, bunda \(E\) \({x^2} + {y^2} + {z^2} = 4\) ning \(x \le 0\) bilan bir qismidir.
6. \( \displaystyle \iiint\limits_{E}{{2 + 16x\,dV}}\)ni baholang, bunda \(E\) \({x^2} + {y^2} + {z^2} = 1\) va \({x^2} + {y^2} + {z^2} = 4\) bilan \(y \ge 0\) va \(z \le 0\) sharlari orasidagi mintaqadir.
7. Avval silindrsimon koordinatalarda integralga aylantirib, quyidagi integralni baholang. \[\int_{0}^{2}{{\int_{{ - \sqrt {4 - {x^{\,2}}} }}^{0}{{\int_{{\sqrt {5{x^{\,2}} + 5{y^{\,2}}} }}^{{\sqrt {9 - {x^{\,2}} - {y^{\,2}}} }}{{\,\,\,7x\,\,\,dz}}\,dy}}\,dx}}\]
8. Avval silindrsimon koordinatalarda integralga aylantirib, quyidagi integralni baholang. \[\int_{{ - \sqrt 5 }}^{{\sqrt 5 }}{{\int_{0}^{{\sqrt {5 - {y^{\,2}}} }}{{\int_{{ - \sqrt {10 - {x^{\,2}} - {y^{\,2}}} }}^{{ - \sqrt {{x^{\,2}} + {y^{\,2}}} }}{{\,\,\,3x{z^2}\,\,\,dz}}\,dx}}\,dy}}\]
9. \(a\) radiusi bo'lgan shar hajmini olish uchun sferik koordinatalarda uch karrali integraldan foydalaning.