computation of moment of spherical shell

In using the formula for area integration over a sphere derived in the example, we need to keep in mind that to every point in xy plane, there correspond two points on the sphere, which are obtained by taking the two signs of the square root. The importance of this fact in obtaining a correct answer is illustrated by our next example, the calculation of the moment of inertia of a spherical shell.

The moment of a spherical shell is given by the integral


While we could compute this by first converting to spherical coordinatesMathworldPlanetmath and then using the result of 1, we can avoid the trouble of changing coordinates by treating the sphere as a graph. Using the result of the previous example, our integral becomes


where the factor of 2 takes into account the observation of the preceding paragraph that two points of the sphere correspond to each point of the xy plane. Computing this integral, we find


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Title computation of moment of spherical shell
Canonical name ComputationOfMomentOfSphericalShell
Date of creation 2013-03-22 14:58:11
Last modified on 2013-03-22 14:58:11
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 13
Author rspuzio (6075)
Entry type Example
Classification msc 28A75