Show that when the density of the region is proportional to the distance from the $y$-axis, the moment of inertia about the y-axis is

$I_y={\int }_1^2{\int }_{-x+2}^{2x-1}k{x}^{3}dydx=\frac{147}{20}k$

The expression is

Plot the vertices $(1,1),(2,0)$, and $(2,3)$and join them.

Obtain the equation of $AB$by using the formula of coordinate geometry

$$\begin{gathered} y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right) \\ y-1=\frac{0-1}{2-1}(x-1) \\ y=-x+2 \end{gathered}$$

Equation of BC

$$\begin{gathered} y-0=\frac{3-0}{2-2}(x-2) \\ y-0=\frac{3}{0}(x-2) \\ x-2=0[\text{Cross multiply]} \\ x=2 \end{gathered}$$

And equation of $CA$is

$y=2x-1$

The moment of inertia of the mass in $\Omega$about the $y$axis is

$I_y={\iint }_{\Omega }{x}^2\rho (x,y)dA$

Where $\rho (x,y)$is the density of the region $\Omega .$

Here $\rho (x,y)$is proportional to the distance from $y$-axis.

Assume $\rho (x,y)=kx.$Then

$I_y={\iint }_{\Omega }{x}^{3}dydx$

Impose the limits on integrals.

$I_y={\int }_1^2{\int }_{-x+2}^{2x-1}k{x}^{3}dydx$

Integrate the inner integral first

$I_y=k{\int }_1^2\left[{\int }_{-x+2}^{2x-1}{x}^{3}dy\right]dx$

Integrate with respect to y

$I_y=k{\int }_1^2{\left[x^{3}y\right]}_{-x+2}^{2x-1}dx$

Substitute the limits

localid="1650645885933" $$\begin{gathered} I_y=k{\int }_1^2{x}^3\left[\right(2x-1)-(-x+2\left)\right]dx{I}_y \\ =k{\int }_1^2\left(3x^4-3x^3\right)dx\text{[Simplify]} \end{gathered}$$

Integrate with respect to $x$

$I_y=k{\left[\frac{3}{5}{x}^5-\frac{3}{4}{x}^4\right]}_1^2$

Substitute the limits

Thus, the moment of inertia about the $y$ axis is

$I_y=\frac{147}{20}k$