We have also discussed the polar moment of inertia in our previous article here we will learn only about the area moment of inertia. It signifies the resistance of an area against the applied moment ( bending moment or twisting moment ) about an axis. The area moment of inertia is a geometrical property of an area that indicates how its points are distributed about an axis. You need to use little wedge shapes pieces.In this article, you will learn a complete overview of the area moment of inertia or second moment of area such as its definition, formula of different sections, units, calculation, and many more. Even as you make them infinitesimally thin this fails. Put another way, take a bunch of long thin rectangles of the same size cut out of paper and try to arrange them into an approximate segment of a circle without overlapping them. But looking at the diagram, clearly surface elements close to the origin need to be smaller than ones further away, otherwise you end up "overcounting" area near $r=0$ and under-counting area further out. So why doesn't your integral using a line work? Well, this treats surface elements at all distances from the origin as having the same size (I'm speaking very loosely here, since infinitesimals don't really have a size). It follows that the width of the surface element is $r\ d\theta$ (the angle shrinks to an infinitesimal, but the radial coordinate does not - it simply takes the value of the radius wherever we place our surface element). It should be easy to convince yourself that the length of an arc that spans angle $theta$ is $r\theta$ (of course $theta$ in radians). But the length of the arc between two evenly spaced blue lines clearly increases as the radial coordinate increases. We don't need to worry about this because in the infinitesimal limit, they approach the same length. First, you might notice that the "inner width" and "outer width" are different. The length is easy, it is $dr$, and is always the same (notice that the length of a blue segment between the evenly spaced red lines is always the same). In the infinitesimal limit the area of one such segment is just its length multiplied by its width. The surface element has the same shape as one of the spaces between two red and two blue lines (a sort of curved rectangle). The reason for this can be seen geometrically: When integrating in (2D) polar coordinates you need to use a surface element:
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