using the method of integration find the area of the region bounded by the lines 2x+y=4, 3x-2y=6 and x-3y+5=0 - Maths - Application of Integrals

Question

using the method of integration find the area of the region
bounded by the lines 2x+y=4, 3x-2y=6 and x-3y+5=0

Aakash Sharma · Accepted Answer

Dear Student! Here is the answer to your query The equations of the given lines are 2x + y = 4 (1) 3x â€“ 2y = 6 (2) x + 3y + 5 = 0 (3) The line 2x + y = 4 meets x and y axis at (2,0) and (0, 4) respectively By Joining these two points we obtain the graph of 2x + y = 4 Similarly the graphs of other two lines are drawn in the figure

Solving equations (1), (2) and (3) in pairs we have $\begin{matrix} 2 x + y = 4 \\ 3 x - 2 y = 6 \end{matrix}} \Rightarrow x = 2 and y = 0 \begin{matrix} 3 x - 2 y = 6 \\ x - 3 y + 5 = 0 \end{matrix}} \Rightarrow x = 4 and y = 3 \begin{matrix} x - 3 y + 5 = 0 \\ 2 x - y = 4 \end{matrix}} \Rightarrow x = 1 and y = 2$ Thus the points of intersection of the given lines are A (2, 0), B (4, 3) and C(1, 2) When we slice the shaded region into vertical strips we observe that the strip change their character at A âˆ´ Required area = (Area CAD) + (Area DAB) $Now Area of Δ CAD = \int_{1}^{2} (y_{2} - y_{1}) ⅆ y = \int_{1}^{2} [\frac{5 + x}{3} - (4 - 2 x)] ⅆ x = \int_{1}^{2} (\frac{7 x}{3} - \frac{7}{3}) ⅆ x = \frac{7}{3} \int_{1}^{2} (x - 1) ⅆ x = \frac{7}{3} [\frac{x^{2}}{2} - x] \begin{matrix} 2 \\ 1 \end{matrix} = \frac{7}{3} [2 - 2 - \frac{1}{2} + 1] = \frac{7}{3} \times \frac{1}{2} = \frac{7}{6} sq units and Area of Δ DAB = \int_{2}^{4} (y_{4} - y_{3}) ⅆ y = \int_{2}^{4} [\frac{5 + x}{3} - (\frac{3}{2} x - 3)] ⅆ x = \int_{2}^{4} [\frac{7 x}{6} + \frac{14}{3}] ⅆ x = \frac{7}{3} \int_{2}^{4} (\frac{x}{2} + 2) ⅆ x = \frac{7}{3} [\frac{- x^{2}}{4} + 2 x] \begin{matrix} 4 \\ 2 \end{matrix} = 0 = \frac{7}{3} [- 4 + 8 + 1 - 4] = \frac{7}{3} sq units ∴ Required area = (\frac{7}{6} + \frac{7}{3}) sq units = \frac{21}{6} sq units = \frac{7}{2} sq units = 3 5 sq uni ts$ Cheers!

using the method of integration.find the area of the region bounded by the lines 2x+y=4, 3x-2y=6 and x-3y+5=0