Final answer:
The sum of the product of each element of an area and the square of its distance from an axis is the moment of inertia, involving integrating over every 'piece of mass' for non-point masses and summing mr² for point masses.
Step-by-step explanation:
The sum of the products of each element of an area and the square of its distance from a coplanar axis of rotation is known as the moment of inertia. The moment of inertia for a system of point particles is calculated by summing or integrating the mass of each point particle multiplied by the square of its distance (r) from the axis of rotation (mr²). For rigid bodies with distributed mass, the calculation of the moment of inertia involves integrating over every 'piece of mass' of the object, considering the square of the distance to the axis of rotation.
For example, the moment of inertia of a single point particle is simply represented by mr², where m is the mass of the particle, and r is the distance from the particle to the axis of rotation. In cases of objects with non-uniform shapes, the calculation is more complex and involves breaking down the object into infinitesimally small pieces of mass (dm) and using integral calculus to find the sum.