# Using integration, find the area of the region bounded by the triangle whose vertices are (-1, 2) ( 1, 5) and ( 3, 4)

Vertices of the triangle are A(-1, 2), B(1,5) and C(3,4)
Equation of the line segemnt AB is
$\large\frac{y-2}{5-2} = \large\frac{x+1}{1+1}$
$\Rightarrow \large\frac{y-2}{3} = \large\frac{x+1}{2}$ i.e., $2(y-2)=3(x+1)$
$2y-4 = 3x+3$
$2y=3x+7$
$y=\large\frac{3x+7}{2}$
Equation of the line segment BC is
$\large\frac{y-5}{4-5} = \large\frac{x-1}{3-1}$
$\large\frac{y-5}{-1} = \large\frac{x-1}{2}$
$y = 5 = \large\frac{-1}{2}$$(x-1) y = \large\frac{-x+11}{2} Equation of the line segment CA is \large\frac{y-4}{2-4} = \large\frac{x-3}{-1-3} \Rightarrow \large\frac{y-4}{-2} = \large\frac{x-1}{-4} \Rightarrow y-4 = \large\frac{1}{2}$$ (x-3)$
$y = \large\frac{x+5}{2}$
edited Apr 1, 2014