91Ó°ÊÓ

The density of the region is constant, the first moment about the $x$-axis is

$M_x={∫}_1^2{∫}_{-x+2}^{2x-1}ydydx=2$

The objective of this problem is to show that when the density of region is constant the first moment about the $x$axis is

$M_x={∫}_1^2{∫}_{-x+2}^{2x-1}ydydx=2$

Plot the vertices localid="1650646891787" $(1,1),(2,0)$, and localid="1650646895001" $(2,3)$and join them.

First moment of the mass in $\mathrm{Î\copyright }$about the $x$axis is

$M_x=âˆ\neg y\mathrm{ÒÏ}(x,y)dA$

Where $\mathrm{ÒÏ}(x,y)$is the density of the region $\mathrm{Î\copyright }$.

Here $\mathrm{ÒÏ}(x,y)$is constant.

Assume $\mathrm{ÒÏ}(x,y)=k$. Then

$M_x=âˆ\neg ykdA$

Impose the limits on integrals.

$M_x={∫}_1^2{∫}_{-x+2}^{2x-1}kydydx$

Integrate the inner integral first

$M_x=k{∫}_1^2\left[{∫}_{-x+2}^{2x-1}ydy\right]dx$

Integrate with respect to $y$

$M_x={∫}_1^2{\left[\frac{y^2}{2}\right]}_{-x+2}^{2x-1}dx[\text{Take}k=1]$

Substitute the limits

$$\begin{gathered} M_x={∫}_1^2\left[\frac{(2x-1{)}^2}{2}-\frac{(-x+1{)}^2}{2}\right]dx \\ {M}_x={∫}_1^2\left[\frac{4x^2-4x+1}{2}-\frac{x^2-2x+1}{2}\right]dx \\ {M}_x=\frac{1}{2}{∫}_1^2\left(3x^2-2x\right)dx[\text{Simplify]} \end{gathered}$$

Integrate with respect to $x$

$M_x=\frac{1}{2}{\left[\frac{3}{3}{x}^3-\frac{2}{2}{x}^2\right]}_1^2$

Substitute the limits

$$\begin{gathered} M_x=\left[(2{)}^3-(2{)}^2-(1{)}^3+(1{)}^2\right] \\ {M}_x=\frac{1}{2}\left[4\right] \\ {M}_x=2[\text{Simplify]} \end{gathered}$$

Thus, the first moment of the mass in $\mathrm{Î\copyright }$about the $x$axis is

$M_x=2$