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Second moment of area

:''This article is about the geometrical property of an area, termed the second moment of area. For the moment of inertia dealing with the rotation of an object with mass, see mass moment of inertia.''
:''For a list, see list of area moments of inertia.''
The second moment of area, also known as moment of inertia of plane area, area moment of inertia, or second area moment, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The second moment of area is typically denoted with either an $I$ for an axis that lies in the plane or with a $J$ for an axis perpendicular to the plane. Its unit of dimension is length to fourth power, L4.
In the field of structural engineering, the second moment of area of the cross-section of a beam is an important property used in the calculation of the beam's deflection and the calculation of stress caused by a moment applied to the beam.
: ''Note: Different disciplines use "Moment of Inertia" (MOI) to refer to either or both of the planar second moment of area, $I = \int_A x^2\, \mathrm dA$, where x is the distance to some reference plane, or the polar second moment of area, $I = \int_A r^2\, \mathrm dA$, where r is the distance to some reference axis. In each case the integral is over all the infinitesimal elements of ''area'', dA, in some two-dimensional cross-section. In math and physics, "Moment of Inertia" is strictly the second moment of mass with respect to distance from an axis: $I = \int_m r^2 \mathrm dm$, where r is the distance to some potential rotation axis, and the integral is over all the infinitesimal elements of ''mass'', dm, in a three-dimensional space occupied by an object. The MOI, in this sense, is the analog of mass for rotational problems.'' In engineering (especially mechanical and civil), "Moment of Inertia" commonly refers to the second moment of the area.〔Beer, Ferdinand (2009). Vector Mechanics for Engineers: Statics. McGraw-Hill. ISBN 978-0-07-352940-0.〕
==Definition==

The second moment of area for an arbitrary shape with respect to an arbitrary axis $BB$ is defined as
: $J_ = \int_A ^2 \, \mathrm dA$
: $\mathrm dA$ = Differential area of the arbitrary shape
: $\rho$ = Distance from the axis BB to dA〔Pilkey, Walter D. (2002). Analysis and Design of Elastic Beams. John Wiley & Sons, Inc. ISBN 0-471-38152-7.〕
For example, when the desired reference axis is the x-axis the second moment of area, $I_$ (often denoted as $I_x$) can be computed in Cartesian coordinates as
: $I_ = \iint_A y^2\, \mathrm dx\, \mathrm dy$
The second moment of the area is crucial in Euler–Bernoulli theory of slender beams.

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