Q. 17

Question

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

$M_{x} = \int_{1}^{2} \int_{- x + 2}^{2 x - 1} k x y d y d x = \frac{27}{8} k$

Step-by-Step Solution

Verified

Answer

Thus, the first moment of the mass in $Ω$ about the $x$ axis is

$M_{x} = \frac{17}{8} k$

1Step 1: Given information

The expression is

M_{x} = \int_{1}^{2} \int_{- x + 2}^{2 x - 1} k x y d y d x = \frac{27}{8} k

2Step 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

$y - y_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} (x - x_{1}) y - 1 = \frac{0 - 1}{2 - 1} (x - 1) y = - x + 2$

Equation of $B C$

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

And equation of $C A$

$y = 2 x - 1$

First moment of the mass in $Ω$ about the $x$ axis is

$M_{x} = \iint y ρ (x, y) d A$

Where $ρ (x, y)$ is the density of the region $Ω$ .

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

Assume $ρ (x, y) = k x$ . Then

$M_{x} = \iint k x y d y d x$

Impose the limits on integrals.

$M_{x} = \int_{1}^{2} \int_{- x + 2}^{2 x - 1} k x y d y d x$

Integrate the inner integral first

$M_{x} = k \int_{1}^{2} [\int_{- x + 2}^{2 x - 1} x y d y] d x$

Integrate with respect to $y$

$M_{x} = k \int_{1}^{2} {[\frac{x y^{2}}{2}]}_{- x + 2}^{2 x - 1} d x$

Substitute the limits

$M_{x} = k \int_{1}^{2} [\frac{x (2 x - 1)^{2}}{2} - \frac{x (- x + 2)^{2}}{2}$

$M_{x} = k \int_{1}^{2} [\frac{4 x^{3} - 4 x^{2} + x}{2} - \frac{x^{3} - 4 x^{2}}{2}$

$M_{x} = \frac{k}{2} \int_{1}^{2} (3 x^{3} - 3 x^{2}) d x [Simplify]$

Integrate with respect to $x$

$M_{x} = \frac{k}{2} {[\frac{3}{4} x^{4} - \frac{3}{3} x^{3}]}_{1}^{2}$

Substitute the limits

$M_{x} = \frac{k}{2} [(\frac{3}{4} (2)^{4} - (2)^{3}) - (\frac{3}{4} - 1)] M_{x} = \frac{17}{8} k [Simplify]$

Thus, the first moment of the mass in $Ω$ about the $x$ axis is

$M_{x} = \frac{17}{8} k$

Q. 16

Q. 18

Other exercises in this chapter

Q. 15

Show that when the density of the region is proportional to the distance from the y-axis, the mass of Ω is given by∫12∫-x+22x-1kxdydx=52k

View solution

Q. 16

Show that when the density of the region is proportional to the distance from the y-axis, the first moment about the y-axis isMy=∫12∫-x+22x-1kx2dydx

View solution

Q. 18

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 isIy=∫12∫-x+22x-1kx

View solution

Q. 19

Show that when the density of the region is proportional to the distance from the y-axis, the first moment about the x-axis isIx=∫12∫-x+22x-1kxy2dyd

View solution