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The quantity of moment of inertia is a very important subject in engineering when calculating the deflection of beams and analyzing distributed forces. In simple words, the moment of inertia is the resistance of a body to rotate, move, or deflect. The greater the moment of inertia, the greater is the resistance. This quantity can be either described in terms of area or in terms of mass. In either case, the rotation of a body occurs about the x-axis or the y-axis. In some instance the rotation can also occur about some angle between the x- and y-axis.
The moment of inertia about the x-axis is mathematically expressed as
Ix = ∫y^2dA
The moment of inertia about the y-axis is expressed as
Iy = ∫x^2dA
When the body rotates about the x- and y-axis at the same time, that is 45º from the x-axis, this quantity is expressed as the product of inertia
Ixy = ∫xydA
If the object rotates about the origin in a circular motion, the quantity is expressed as the polar moment of inertia.
Io = ∫r^2dA
The distances of x,y, and r are taken from the applied force to the rotational axis. This is when the concept of centroids has to be considered. Instead of computing each individual point of the area, the integral is the sum of all points within a boundary.