How do calculate this moment of inertia?

[LCSphysicist]

Homework Statement

Doubt conceptual about the inertia moment of a thin square

Relevant Equations

I = int dm r square

Inertia moment of a thin square side 2b about the center of mass...

I put the coordinates in the center of the square and came to:

Integral of (x²+y²)dm
=
Integral of (x²+y²)*(dxdy)M
But, the interval of the integral is [0,b] to x and y
And, since this consider just the integral of one quadrant, multiple by 4 to the total square

I get 8mb²/3 , which is four times more than the answer, 2mb²/3.
Where did i wrong?

 

[haruspex]

Homework Statement:: Doubt conceptual about the inertia moment of a thin square
Relevant Equations:: I = int dm r square

Inertia moment of a thin square side 2b about the center of mass...

I put the coordinates in the center of the square and came to:

Integral of (x²+y²)dm
=
Integral of (x²+y²)*(dxdy)M
But, the interval of the integral is [0,b] to x and y
And, since this consider just the integral of one quadrant, multiple by 4 to the total square

I get 8mb²/3 , which is four times more than the answer, 2mb²/3.
Where did i wrong?

Is m the mass of the square or of one quadrant?

 

[LCSphysicist]

Is m the mass of the square or of one quadrant?

Omg thx

 

[haruspex]

Omg thx

I should point out that you only specified the axis as being through the centre of mass. You did not say whether it is parallel to a side of the square, but perhaps assumed it is.
Interestingly, it doesn't matter!
Have you come across the https://en.m.wikipedia.org/wiki/Perpendicular_axis_theorem?