Using the method of integration find the area of the region bounded by lines:

The given equations of lines are

2x + y = 4 … (1)

3x – 2y = 6 … (2)

And, x – 3y + 5 = 0 … (3)

The area of the region bounded by the lines is the area of ΔABC. AL and CM are the perpendiculars on x-axis.

Area (ΔABC) = Area (ALMCA) – Area (ALB) – Area (CMB)

$=\int_{1}^{1}\left(\frac{x+5}{3}\right) d x-\int_{1}^{2}(4-2 x) d x-\int_{2}^{4}\left(\frac{3 x-6}{2}\right) d x$

$=\frac{1}{3}\left[\frac{x^{2}}{2}+5 x\right]_{1}^{4}-\left[4 x-x^{2}\right]_{1}^{2}-\frac{1}{2}\left[\frac{3 x^{2}}{2}-6 x\right]_{2}^{4}$

$=\frac{1}{3}\left[8+20-\frac{1}{2}-5\right]-[8-4-4+1]-\frac{1}{2}[24-24-6+12]$

$=\left(\frac{1}{3} \times \frac{45}{2}\right)-(1)-\frac{1}{2}(6)$

$=\frac{15}{2}-1-3$

$=\frac{15}{2}-4=\frac{15-8}{2}=\frac{7}{2}$ units