Proof for moment of Inertia of thin disc

[Redoctober]

Homework Statement

Its actually not a homework , i am just curious about this

I have disc with a hole in middle , R1 is inner radius , R2 is outer radius , Mass M is the disc's mass . Let density of disc uniform .

I did it this way . -

λ.dA=dm
λ.pie.r.dr=dm as Area=pie.r^2

Let substute in the equation I = ∫( R^2).dm

therefore I = ∫( p.pie.r^3 ).dr
remove constants outside the integral therefore I =λ.pie∫r^3.dr
integrate (0 to R )to get I = λ.pie.r^4/4
using Mass = λ*pie*r^2

therefore i get I = (MR^2)/4 for thin Disc

As i have a hole therefore its I = M(R2^2-R1^2)/4

But sadly this is incorrent :/ . It seems that I = M(R2^2-R1^2)/2 is the correct solution

Please Help ! :O why my way is wrong :(

 

[Spinnor]

you wrote,

λ.pie.r.dr=dm

Shouldn't there be a 2 in there? dA = 2.pie.r.dr ?

 

[Redoctober]

you wrote,

λ.pie.r.dr=dm

Shouldn't there be a 2 in there? dA = 2.pie.r.dr ?

Oh my god "Slap myself in face" Lol such a silly error :S ! .