Q. 16 - Calculus

Question

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

$M_y={\int }_1^2{\int }_{-x+2}^{2x-1}k{x}^{2}dydx=\frac{17}{4}k$

Step-by-Step Solution

Answer

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

$M_y=\frac{17}{4}k$

Step 1: Given information

Th expression is

$M_y={\int }_1^2{\int }_{-x+2}^{2x-1}k{x}^{2}dydx=\frac{17}{4}k$

Step 2: Calculation

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

Obtain the equation of $A B$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 $B C$

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

And equation of$C A$

$y=2 x-1$

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

$M_y=\iint x\rho (x,y)dA$

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

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

$$\begin{gathered} \rho (x,y)=xk.\text{ Then } \\ {M}_y=\iint k{x}^{2}dA \end{gathered}$$

Impose the limits on integrals.

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

Integrate the inner integral first

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

Integrate with respect to $$y$$

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

Substitute the limits

$$\begin{gathered} M_y=k{\int }_1^2\left[\right(2x-1)-(-x+2\left)\right]x^{2}dx \\ {M}_y=k{\int }_1^2\left[\right(2x-1)-(-x+2\left)\right]x^{2}dx \\ {M}_y=k{\int }_1^2\left(3x^3-3x^2\right)dx[\text{ Simplify]} \end{gathered}$$

Integrate with respect to $x$

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

Substitute the limits

$$\begin{gathered} M_y=k\left[\left(\frac{3}{4}(2{)}^4-(2{)}^3\right)-\left(\frac{3}{4}-1\right)\right] \\ {M}_y=\left[(12-8)+\frac{1}{4}\right] \\ {M}_y=\frac{17}{4}k[\text{ Simplify]} \end{gathered}$$

Thus, the first moment of the mass in $\Omega$ about the $y$ axis is

$M_y=\frac{17}{4}k$