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## Homework Statement

Derive the moment of inertia of a solid, uniform density, disk rotating about a diameter using vertical rods.

Diagram: http://i.imgur.com/TQIjz.png

Only the radius of the circle and the inertia of a thin rod is given

## Homework Equations

Inertia of thin vertical rod = m*r^2

σ= area density

M= Total mass

I= inertia

dm= infinitesimal amount of mass

dI= infinitesimal amount of inertia

A= total area

R= radius of the circle

## The Attempt at a Solution

This is the method given in my class. Because calculus 1 is the only prerequisite we can not use double integrals.

I of rod = m*x^2

dI= dm

*x^2dm=σ*da

da= 2y*dx

σ= M/A

A=pi*R^2

σ= M/(pi*R^2)

dm=M/(pi*R^2) * 2y*dx

dI= M/(pi*R^2) * 2y*x^2 * dx

From here we have to put y in terms of x which I think must be done using the Pythagorean theorem so:

y=(R^2-x^2)^(1/2)

subbing in we get:

dI= M/(pi*R^2) * 2(R^2-x^2)^(1/2) * x^2 * dx

This is why I get stuck because I am not quite sure how to integrate that. What is should come out to be is I = 1/4 MR^2. If anyone can offer an explanation it would be very appreciated.