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Homework Statement
I was trying to find the moment of inertia of a rectangle with width a and height b were axis of rotation is through it's center of mass
Homework Equations
The Attempt at a Solution
I = integral r^2 dm
rho = dm/dA
dm = rho dA
If I take an infinitely small area of an infinitely thin rectangle a distance dr from the point of rotation, it would have a width of dr and a height of b, hence
dA = b dr
dm = rho b dr
I = rho b integral[-a/2,a/2] r^2 dr
I = (rho b)/3 * r^3 |[-a/2,a/2]
I = (rho b)/3 * [ a^3 /8 - (- a^3/8) ]
I = (rho b)/3 * [ a^3/8 + a^3 / 8]
I = (rho b a^3)/12
rho = M/A
A = ab
I = (M b a^3)/(12 a b)
I = (M a^2)/12
I don't see what I'm doing wrong, my book tells me it's I = [ M(a^2 + b^2) ] /12
thanks for any help