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The moment of inertia of a solid about the \( z \)-axis is

  • A.\( \displaystyle\iiint \left(x^{2}+y^{2}+z^{2}\right)\rho\,dV \)
  • B.\( \displaystyle\iiint z^{2}\rho\,dV \)
  • C.\( \displaystyle\iiint \sqrt{x^{2}+y^{2}}\,\rho\,dV \)
  • D.\( \displaystyle\iiint \left(x^{2}+y^{2}\right)\rho\,dV \)

Answer: D

The moment of inertia weights each element by the square of its distance from the axis, and the distance from the \( z \)-axis is \( \sqrt{x^{2}+y^{2}} \), whose square is \( x^{2}+y^{2} \). Including \( z^{2} \) would give the polar moment about the origin, and using the first power of the distance would give a first moment rather than a second

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