1. A uniform thin disk has radius “R” and mass “M”
A uniform thin disk has radius “R” and mass “M”. Find the moment of inertia of the disk about an axis tangent to the disk.
axis of rotation
Request: try to solve the problem without using references.
Hint: there are smart ways to solve the problem, and there are ‘conventional’ methods [i.e. through integrating r2 dm].
Suggestion: solve the problem BOTH ways. Of course, you should get the same result both ways!  R M

2. axis of rotation y1 y R O x1 M :z1 First, the smart way:
1- We all know that the moment of inertia of a uniform disk (M,R) about an axis (z1) through its center (O) perpendicular to the plane of the disk [see red dot in the figure] is Iz1 = ½ M R2.  one can easily integrate to find this out, but all physicists probably memorize it by heart!
2- You should also know that for any planar body, the moment of inertia about an axis perpendicular to the plane of the body is equal to the sum of the moments of inertia about any two orthogonal axes in the plane of the body where the three axes intersect  this is sometimes called the perpendicular axis theorem (it applies only for planar bodies): Hence:
Ix1 + Iy1 = Iz1
3- From symmetry in our situation, Ix1= Iy1.
Therefore,
2 Iy1 = ½ M R2
Iy1 = ¼ M R2
4- Noticing that O is at the center of mass use the parallel axis theorem:
Iy = Iy1 + M D2 = Iy1 + M R2
Iy = 5/4 M R2

3. Now, the direct integration method with the help of Mathematica
axis of rotation x y
The red strip has an area da = height times width
The height is (2 y) and the width is dx

4. Now, the direct integration method without the help of Mathematica 
axis of rotation x y