ES-210 Moment of Inertia by Integration
The moment of inertia of a body about the x-axis is defined by

where the element of area is taken parallel to the x-axis.

The moment of inertia about the y-axis is defined by

where the element of area is taken parallel to the y-axis.


The moment of inertia of a rectangle about its edge
is given by



The moments of inertia Ixx and Iyy may be calculated from the same element by



where dIxx is the moment of inertia of a rectangle of infinitesimal base b=dx and height h=y,
and dIyy is the moment of inertia of a thin rectangle about the y axis.

The polar moment of inertia of an object is defined by
  
where the geometric configuration is as shown...


    
The polar moment of inertia may be expressed in terms of the rectangular moments of inertia as...



The radius of gyration (k) is defined as the location where a concentrated area should be placed so as to have the same moment of inertia and area as the original body


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