Question

1. the moment if inertia of an area is defined as the sum of the products of all elementary areas multiplied by the square of their distances from an axis. where; ( i_{x - x}=) (az 2 ( a=) elementary area ( x - x=) axis of rotation ( z=) longitudinal axis of the beam

Explanation:

Step1: Recall the formula for moment of inertia

The moment of inertia \(I_{X - X}\) of an area about an axis \(X - X\) is given by the integral formula \(I_{X - X}=\int_{A}z^{2}da\), where \(da\) is the elementary area, \(X - X\) is the axis of rotation and \(z\) is the perpendicular distance from the elementary area \(da\) to the axis \(X - X\). In the discrete form (if we consider summing up small areas \(\Delta a\) instead of integrating), \(I_{X - X}=\sum a z^{2}\)

Answer:

\(I_{X - X}=\sum a z^{2}\)