Question

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

Answer 1

first we have to find out the equation of lines

so $L_1,L_2,L_3$ are $3x-2y+7=0$, $x-2y+5=0$, $x+2y-11=0$ respectively

line $L_1$ and $L_2$ cut each other at $(-1,2)$

similarly $L_2$ and $L_3$ cut each other at $(3,4)$

and $L_1$ and $L_3$ cut each other at $(1,5)$

$A={\int }_{-1}^1(\frac{3x}{2}+\frac{7}{2})dx+{\int }_1^3(\frac{11}{2}-\frac{x}{2})dx-{\int }_{-1}^3(\frac{x}{2}+\frac{5}{2})dx$

on integrating the above equation and putting the upper and lower limit ,we will get

$A=4$