Remember that kinetic energy is ½mv2. Consider a small piece of the object that is a distance r from the z axis. Its velocity, in meters per second, is 2πkr. Square this and multiply by ½f to get the energy. Therefore the total energy stored in the spinning object is the integral of 2(πkr)2f. This is the second moment in the x and y directions, also called the second moment about z.
It is usually convenient to integrate in polar or cylindrical coordinates, which brings in another factor of r. If f is a uniform disk or cylinder, its density function is 1. Integrate 2π2k2r3 throughout the disk of radius a and get π3k2a4.
The factor k2 is often pulled out, leaving the angular inertia. This is a measure of how hard it is to start the object spinning, or to stop it once it is spinning. Multiply the inertia by k2 to get the angular momentum.
A fly wheel is designed to maximize angular momentum. The majority of the mass is placed at the outer ring, where it is moving the fastest. This looks something like a bicycle wheel, except the tire is made of solid metal. For all practical purposes, the angular inertia of a fly wheel is 2π2r2M, where m is the mass of the outer ring.
Let's find the angular inertia of a sphere of radius a. This might be used to determine the energy stored in a spinning planet.
The distance from the z axis is r×sin(φ). This is squared, and we need another factor of r×sin(φ) for the privilege of using spherical coordinates. Thus the integrand becomes 2π2(r×sin(φ)3. Sine3 is the same as sine - sine×cosine2, and these two terms are easily integrated, giving -cosine + cosine3/3. Evaluate from 0 to π and get 4/3. Integrate with respect to r and θ and the angular inertia is 4π3a4/3. Multiply by k2 to get the angular momentum in a spinning sphere.