Derive an expression for moment of inertia of a rectangular plate of sides a and b, where a is the longer side; about an axis parallel to width b and passing through the centre.

what will be the moment of inertia about an axis perpendicular to the plane of the plate passing through its centre c?

(a) Moment of inertia of a rectangular plate of sides a and b, where a is the longer side; about an axis parallel to width b and passing through the centre-
Take a rectangular plate having length a and width b.Now draw a line AB passing through centre parallel to b.Draw a line parallel to AB at a distance x from it and another at a distance x+dx.We can take a strip enclosed between the two lines as the small element.
As plate is uniform,
Its mass per unit area = $\frac{M}{ba}$
Mass of the strip = $\frac{M}{ba}b dx = \frac{M}{a}dx$
The perpendicular distance of the strip from AB = x .
The moment of inertia of the strip about AB = dI = $\frac{M}{a}dx x^2$
$$\begin{gathered} Therefore, \\ I = {\int }_{-a/2}^{a/2}\frac{M}{a}{x}^2 dx =\frac{ M{a}^2}{12} \end{gathered}$$

 Moment of inertia about an axis perpendicular to the plane of the plate passing through its centre c-
According to the  theorem of perpendicular axis, which is

Iz = Ix + Iy

here

Ix =$\frac{M{b}^2}{12}$

so,

Iy =$\frac{M{a}^2}{12}$

So,Iz = Ix + Iy
Iz = $$\begin{gathered} =\frac{M{a}^2}{12}+\frac{M{b}^2}{12} \\ = \frac{M(a^2+b^2)}{12} \end{gathered}$$