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Section 4-6 : Triple Integrals in Cylindrical Coordinates

  1. Evaluate \( \displaystyle \iiint\limits_{E}{{8z\,dV}}\) where \(E\) is the region bounded by \(z = 2{x^2} + 2{y^2} - 4\) and \(z = 5 - {x^2} - {y^2}\) in the 1st octant.
  2. Evaluate \( \displaystyle \iiint\limits_{E}{{6xy\,dV}}\) where \(E\) is the region above \(z = 2x - 10\), below \(z = 2\) and inside the cylinder \({x^2} + {z^2} = 4\).
  3. Evaluate \( \displaystyle \iiint\limits_{E}{{9y{z^3}\,dV}}\) where \(E\) is the region between \(x = - \sqrt {9{y^2} + 9{z^2}} \) and \(x = \sqrt {{y^2} + {z^2}} \) inside the cylinder \({y^2} + {z^2} = 1\).
  4. Evaluate \( \displaystyle \iiint\limits_{E}{{x + 2\,dV}}\) where \(E\) is the region bounded by \(x = 18 - 4{y^2} - 4{z^2}\) and \(x = 2\) with \(z \ge 0\).
  5. Evaluate \( \displaystyle \iiint\limits_{E}{{x + 2\,dV}}\) where \(E\) is the region between the two planes \(2x + y + z = 6\) and \(6x + 3y + 3z = 12\) inside the cylinder \({x^2} + {z^2} = 16\).
  6. Evaluate \( \displaystyle \iiint\limits_{E}{{{x^2}\,dV}}\) where \(E\) is the region bounded by \(y = {x^2} + {z^2} - 4\) and \(y = 8 - 5{x^2} - 5{z^2}\) with \(x \le 0\).
  7. Use a triple integral to determine the volume of the region bounded by \(z = \sqrt {{x^2} + {y^2}} \), and \(z = {x^2} + {y^2}\) in the 1st octant.
  8. Use a triple integral to determine the volume of the region bounded by \(y = \sqrt {9{x^2} + 9{z^2}} \), and \(y = - 3{x^2} - 3{z^2}\) in the 1st octant.
  9. Use a triple integral to determine the volume of the region below \(x = z + 3\), above \(x = - z - 6\) and inside the cylinder \({y^2} + {z^2} = 4\).
  10. Evaluate the following integral by first converting to an integral in cylindrical coordinates. \[\int_{{ - 4}}^{4}{{\int_{0}^{{\sqrt {16 - {y^{\,2}}} }}{{\int_{0}^{{6 + x}}{{\,\,\,6y{x^2}\,\,dz}}\,dx}}\,dy}}\]
  11. Evaluate the following integral by first converting to an integral in cylindrical coordinates. \[\int_{0}^{3}{{\int_{{ - \sqrt {9 - {x^{\,2}}} }}^{{\sqrt {9 - {x^{\,2}}} }}{{\int_{{ - \sqrt {2{x^{\,2}} + 2{y^{\,2}}} }}^{{6 + {x^{\,2}} + {y^{\,2}}}}{{\,\,\,15z\,\,dz}}\,dy}}\,dx}}\]
  12. Use a triple integral in cylindrical coordinates to derive the volume of a cylinder of height \(h\) and radius \(a\).