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Section 4-7 : Triple Integrals in Spherical Coordinates

  1. Evaluate \( \displaystyle \iiint\limits_{E}{{4{y^2}\,dV}}\) where \(E\) is the sphere \({x^2} + {y^2} + {z^2} = 9\).
  2. Evaluate \( \displaystyle \iiint\limits_{E}{{3x - 2y\,dV}}\) where \(E\) is the region between the spheres \({x^2} + {y^2} + {z^2} = 1\) and \({x^2} + {y^2} + {z^2} = 4\) with \(z \le 0\).
  3. Evaluate \( \displaystyle \iiint\limits_{E}{{2yz\,dV}}\) where \(E\) is the region below \({x^2} + {y^2} + {z^2} = 16\) and inside \(z = \sqrt {3{x^2} + 3{y^2}} \) that is in the 1st octant.
  4. Evaluate \( \displaystyle \iiint\limits_{E}{{{z^2}\,dV}}\) where \(E\) is the region between the spheres \({x^2} + {y^2} + {z^2} = 4\) and \({x^2} + {y^2} + {z^2} = 25\) and inside \(\displaystyle z = - \sqrt {\frac{1}{3}{x^2} + \frac{1}{3}{y^2}} \).
  5. Evaluate \( \displaystyle \iiint\limits_{E}{{5{y^2}\,dV}}\) where \(E\) is the portion of \({x^2} + {y^2} + {z^2} = 4\) with \(x \le 0\).
  6. Evaluate \( \displaystyle \iiint\limits_{E}{{2 + 16x\,dV}}\) where \(E\) is the region between the spheres \({x^2} + {y^2} + {z^2} = 1\) and \({x^2} + {y^2} + {z^2} = 4\) with \(y \ge 0\) and \(z \le 0\).
  7. Evaluate the following integral by first converting to an integral in cylindrical coordinates. \[\int_{0}^{2}{{\int_{{ - \sqrt {4 - {x^{\,2}}} }}^{0}{{\int_{{\sqrt {5{x^{\,2}} + 5{y^{\,2}}} }}^{{\sqrt {9 - {x^{\,2}} - {y^{\,2}}} }}{{\,\,\,7x\,\,\,dz}}\,dy}}\,dx}}\]
  8. Evaluate the following integral by first converting to an integral in cylindrical coordinates. \[\int_{{ - \sqrt 5 }}^{{\sqrt 5 }}{{\int_{0}^{{\sqrt {5 - {y^{\,2}}} }}{{\int_{{ - \sqrt {10 - {x^{\,2}} - {y^{\,2}}} }}^{{ - \sqrt {{x^{\,2}} + {y^{\,2}}} }}{{\,\,\,3x{z^2}\,\,\,dz}}\,dx}}\,dy}}\]
  9. Use a triple integral in spherical coordinates to derive the volume of a sphere with radius \(a\).