Using integration find the area of the triangular region whose sides have the equations y = 2 x +1, y = 3 x + 1 and x = 4.
We have to find the area bounded by the triangle whose sides are given as y = 2 x + 1 , y = 3 x + 1 and x = 4 .
Figure (1)
The area of the triangle ABC is to be calculated.
To find the area of the region OABLO, assume a vertical strip of infinitely small width and integrate it.
Area of the region OABLO = ∫ 0 4 y d x = ∫ 0 4 ( 3 x + 1 ) d x = [ 3 x 2 2 + x ] 0 4 = [ 3 ( 4 ) 2 2 + 4 − ( 3 ( 0 ) 2 2 + 0 ) ]
Simplify further,
Area of the region OABLO = [ 3 ( 4 ) 2 2 + 4 − ( 3 ( 0 ) 2 2 + 0 ) ] 0 4 = 24 + 4 = 28 sq units
Similarly find the area of the region OACLO, assume a vertical strip of infinitely small width and integrate it.
Area of the region OACLO = ∫ 0 4 y d x = ∫ 0 4 ( 2 x + 1 ) d x = [ 2 x 2 2 + x ] 0 4 = [ ( 4 ) 2 + 4 − ( 0 2 + 0 ) ]
Simplify further,
Area of the region OACLO = [ ( 4 ) 2 + 4 − ( 0 2 + 0 ) ] = 16 + 4 = 20 sq units
The area of the shaded triangular region is,
Area of the triangle ABC = Area of the region OABLO − Area of the region OACLO = 28 − 20 = 8 sq units
Thus, the required area of the triangular region is 8 sq units .
We have to find the area bounded by the triangle whose sides are given as
Figure (1)
The area of the triangle ABC is to be calculated.
To find the area of the region OABLO, assume a vertical strip of infinitely small width and integrate it.
Simplify further,
Similarly find the area of the region OACLO, assume a vertical strip of infinitely small width and integrate it.
Simplify further,
The area of the shaded triangular region is,
Thus, the required area of the triangular region is
