The formula for proving $\tau = I\alpha$ for continuous mass distribution is $\tau = \int r \times dm$, where $\tau$ represents the torque, $I$ represents the moment of inertia, and $\alpha$ represents the angular acceleration.
The moment of inertia for a continuous mass distribution is calculated by taking the sum of the products of the mass and the square of the distance from the axis of rotation for each infinitesimal element of the mass distribution, and integrating over the entire mass distribution.
A continuous mass distribution is one in which the mass is distributed continuously throughout a given volume or area, while a discrete mass distribution is one in which the mass is concentrated at specific points. In other words, a continuous mass distribution has an infinite number of infinitesimal elements, while a discrete mass distribution has a finite number of discrete elements.
To prove $\tau = I\alpha$ for continuous mass distribution using calculus, you would need to integrate the formula $\tau = \int r \times dm$ over the entire mass distribution, taking into account the infinitesimal elements of the mass distribution and their respective distances from the axis of rotation. This would result in an expression for the torque in terms of the moment of inertia and the angular acceleration.
Yes, the formula $\tau = I\alpha$ is a general formula that can be applied to any shape or type of mass distribution, as long as the mass distribution is continuous. However, the calculation of the moment of inertia may vary depending on the shape and type of mass distribution, but the principle of integrating over infinitesimal elements remains the same.