Question

Compute the area bounded by the lines x + 2y = 2, y – x = 1 and 2x + y = 7.

Answer 1

Given that: x + 2y = 2  .....(i)

y – x = 1   ......(ii)

And 2x + y = 7   ......(iii)

x	0	2
y	1	0

 

x	0	–1
y	1	0

 

x	0	`7/2`
y	7	0

Solving equations (ii) and (iii)

We get y = 1 + x

∴ 2x + 1 + x = 7

3x = 6

⇒ x = 2

∴ y = 1 + 2

= 3

Coordinates of B = (2, 3)

Solving equations (i) and (iii)

We get x + 2y = 2

∴ x = 2 – 2y

2x + y = 7

2(2 – 2y) + y = 7

⇒ 4 – 4y + y = 7

⇒ –3y = 3

∴ y = –1 and x = 4

∴ Coordinates of C = (4, – 1) and coordinates of A = (0, 1).

Taking the limits on y-axis, we get

`int_(-1)^3 x_"BC" "d"y - int_(-1)^1  x_"AC" "d"y - int_1^3  x_"AB" "d"y`

= `int_(-1)^3 (7 - y)/2  "d"y - int_(-1)^1 (2 - 2y)  "d"y - int_1^3 (y - 1) "d"y`

= `1/2 [7y - y^2/2]_-1^2 - 2[y - y^2/2]_-1^1 - [y^2/2 - y]_1^3`

= `1/2[(21 - 9/2) - (7 - 1/2)] - 2[(1 - 1/2) - (-1 - 1/2)] - [(9/2 - 3) - (1/2 - 1)]`

= `1/2[33/2 + 15/2] - 2[1/2 + 3/2] - [3/2 + 1/2]`

= `1/2 xx 24 - 2 xx 2 - 2`

⇒ 12 – 4 – 2 = 6 sq.units

Hence, the required area = 6 sq.units