Using integration, find the area of `DeltaABC`, whose vertices are `A(2,3), B(4,7)` and `(6,2)`.

Explanation

The equation of AB is
`(y-3)/(x-2)=(7-3)/(4-2)rArr y=2 x-1 `
The equation of BC is `(y-7)/(x-2)=(2-3)/(6-2) rArr y= (-1)/(4)x+7/2`
Draw `AL_|_ Ox ,BM _|_ OX and CN_|_ OX`
Area of `Delta` ABC=(area ALMBA+ are BMNCB)-(area ALNCA)
`=underset(2)overset(4)int y AB dx + underset(4)overset(6)int y aBC dx-underset(2)overset(6)int y ACdx `
`=underset(2)overset(4)int(2x-1)dx+underset(4)overset(6)int(-5/2x+17)dx-underset(2)overset(6)(-1/4x+7/2)dx`
`=[x^2x]_2^4+[-5/4x^2+17 x ]_4^6-[-1/8x^2+7/2x]_2^6`
`=[(12-2)+(57 -48)-(33/2-13/2)]`sq units
= 9 sq units
Hence the required area is 9 sq units .