But if you make it 3D, you have a new moment of inertia:
Is there any connection between these two?
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The general idea in both cases is similar.
In the first case, we are basically summing the differential areas multiplied by the square of their distance from the axis of rotation, neutral axis. This is tightly related to the concept of the linear relationship of stress and strain in the flexural properties of beams and Young's modulus. This is usually called the second area moment or second-moment of inertia.
$$ I_x=\int y^2dA $$
In the second case, as shown in your second diagram, We are summing the differential mass elements of a solid (doesn't have to be a solid, liquid, gas or plasma is fine) again multiplied by the square of their distances from the axis of rotation, to get the inertia for angular momentum and the concepts of rotational energy, work, and preservation of angular momentum and the entire spectrum of related physics. $$ I=\int r^2dm$$