Using integration find the area of the triangular region whose sides have the equations`y = 2x + 1`, `y = 3x + 1`and `x = 4`.

Explanation

The given lines are y = 2x + 1
(i) y = 3x + 1 (ii) x = 4

For intersection point of (i) and (iii) `y = 2xx4 + 1 = 9` Coordinates of intersecting point of (i) and (iii) is `(4, 9)` For intersection point of (ii) and (iii) `y = 3xx4 + 1 = 13`
i.e., Coordinates of intersection point of (ii) and (iii) is `(4, 13)`
For intersection point of (i) and (ii) `2x + 1 = 3x + 1 => x = 0 y = 1`
i.e., Coordinates of intersection point of (i) and (ii) is (0, 1).
Shaded region is required triangular region.
Required Area = Area of trapezium OABD - Area of trapezium OACD
=`int_0^4(3x+1)dx-int_0^4(2x+1)dx`
=`[[(3(x)^2))/2+x]_0^4-[2x^2/2+x]_0^4`
`[24+4-0]-[16+4-0]`=`28-20`
`8`sq units