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Section 4-6 : Triple Integrals in Cylindrical Coordinates

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