A circular lamina of radius `a` and centre `O` has a mass per unit area of `kx^(2)`, where `x` is the distance from `O` and `k` is a constant. If the mass of the lamina is `M`, find in terms of `M` and `a`, the moment of inertia of the lamina about an axis through `O` and perpendicular to the lamina.

Explanation

To find the moment of inertia \( I \) of a circular lamina of radius \( a \) and mass per unit area given by \( kx^2 \), we can follow these steps:

### Step 1: Define the mass element
The mass per unit area of the lamina is given by \( \sigma = kx^2 \), where \( x \) is the distance from the center \( O \). The area element in polar coordinates is given by \( dA = r \, dr \, d\theta \).

### Step 2: Express the mass element
The mass element \( dm \) can be expressed as:
\[
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