Why Does the 1/3 Factor Appear in the Moment of Inertia Calculation?

[salman213]

Ok I am try to understand how to find the moment of inertia of different shapes by using direct integration. In a lot of the solutions I see they always have this one thing that I don't understand. Example:

Determine by direct integration the moment of inertia of the shaded area
with respect to the y axis.

http://img406.imageshack.us/img406/4816/64329189gz9.jpg


WHERE DOES THE 1/3 COME FROM??

I know the formula is

Iy = INTEGRALOF(x^2 dA)

doesnt da = xdy

so that makes it

Iy =INTEGRALOF(x^2 * x *dy)

 

[alphysicist]

Hi salman213,

It appears to me that the way they are setting up the integral is by dividing the area into infinitesimal strips of length x and height dy. Each of these strips has the form of a thin rod which is rotating about one end, and that is where the (1/3) comes from. (The moment of inertia of a thin rod rotating about one end is \frac{1}{3}ML^2.)

The integral then sums up the moments of inertia of these strips.