Compute, using integration, the area bounded by the lines x+2y = 2, y-x=1 and 2x+y= 7

Given: Lines are:

x + 2y = 2 …(i)

y – x = 1 …(ii)

and 2x + y = 7 …(iii)

Adding eq. (i) and (ii), we get

x + 2y + y – x = 2 + 1

⇒ 3y = 3

⇒ y = 1

Substituting the value of y in eq. (ii), we get

1 – x = 1

⇒ x = 0

So, the point of intersection is (0, 1)

Subtracting eq. (ii) from (iii), we get

2x + y – (y – x) = 7 – 1

⇒ 2x + y – y + x = 6

⇒ 3x = 6

⇒ x = 2

Substituting the value of x in eq. (iii), we get

2(2) + y = 7

⇒ 4 + y = 7

⇒ y = 7 – 4

⇒ y = 3

So, the point of intersection is (2, 3)

Solving eq. (i) and (iii)

Multiply eq. (i) by 2 and subtract it from eq. (iii), we get

2x + y – 2(x + 2y) = 7 – 2(2)

⇒ 2x + y – 2x + - 4y = 7 – 4

⇒ -3y = 3

⇒ y = -1

Substituting the value of y in eq. (i), we get

x + 2(-1) = 2

⇒ x – 2 = 2

⇒ x = 4

So, the point of intersection is (4, -1)

Now, Area of shaded region

= 3 + 3

= 6 sq. units