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Section 4-7 : Triple Integrals in Spherical Coordinates

  1. Evaluate \( \displaystyle \iiint\limits_{E}{{10xz + 3\,dV}}\) where \(E\) is the region portion of \({x^2} + {y^2} + {z^2} = 16\) with \(z \ge 0\). Solution
  2. Evaluate \( \displaystyle \iiint\limits_{E}{{{x^2} + {y^2}\,dV}}\) where \(E\) is the region portion of \({x^2} + {y^2} + {z^2} = 4\) with \(y \ge 0\). Solution
  3. Evaluate \( \displaystyle \iiint\limits_{E}{{3z\,dV}}\) where \(E\) is the region below \({x^2} + {y^2} + {z^2} = 1\) and inside \(z = \sqrt {{x^2} + {y^2}} \). Solution
  4. Evaluate \( \displaystyle \iiint\limits_{E}{{{x^2}\,dV}}\) where \(E\) is the region above \({x^2} + {y^2} + {z^2} = 36\) and inside \(z = - \sqrt {3{x^2} + 3{y^2}} \). Solution
  5. Evaluate the following integral by first converting to an integral in spherical coordinates. \[\int_{{ - 1}}^{0}{{\int_{{ - \sqrt {1 - {x^{\,2}}} }}^{{\sqrt {1 - {x^{\,2}}} }}{{\int_{{\sqrt {6{x^{\,2}} + 6{y^{\,2}}} }}^{{\sqrt {7 - {x^{\,2}} - {y^{\,2}}} }}{{\,\,\,18y\,\,\,dz}}\,dy}}\,dx}}\] Solution