Moment of Inertia for a Filled Sphere

[wil3]

Hello. I am attempting to derive the equation for the moment of inertia of a filled sphere. I do not how how to do the proper notation by typing, so I have attached a scan of my work(I apologize if opening the attachment is inconvenient)

I keep getting the correct order final equation, but my coefficient is 3/5 instead of 2/5. I am trying to do this using only single-variable calculus, so please do not use double integrals, etc unless you think they can be explained to a high schooler. If it is actually impossible to solve this problem using only one-variable integration, please tell me so and explain why. Thank you very much.

 

[mathman]

Your note is very hard to read. However from past observations of questions like yours, it usually boils down to a simple error in arithmetic that has been overlooked. I suggest you go over your work carefully, line by line.

 

[gabbagabbahey]

The problem is that the "r" in your moment of inertia definition represents the perpendicular distance from the axis of rotation to the mass element "dm". This is not the same thing as the distance from the origin. You really need to use multivariable calculus to account for this.