Question

Using integration , find the of region bounded by the triangle whose vertices are $A\left(1,0\right),B\left(2,2\right)$ and $C\left(3,0\right)$

Answer 1

Given $A(1,0),B(2,2),C(3,1)$

Equation of line passing through $A(1,0),B(2,2):$

Slope of Line $\frac{2-0}{2-1}=2$

Equation $y-0=2(x-1)2x-y-2=0⟹y=2x-2\cdots \left(1\right)$

Equation of line passing through $C(3,0),B(2,2):$

Slope of Line $\frac{2-0}{2-3}=-2$

Equation $y-0=-2(x-3)2x+y-6=0⟹y=6-2x\cdots \left(2\right)$

The area of triangle is area under $\left(1\right)$ at the limits of $x=1$ to $x=2$

and area under $\left(2\right)$ at the limits of $x=2$ to $x=3$

Area of triangle ${\int }_1^2(2x-2)dx+{\int }_2^3(6-2x)dx{[x^2-2x]}_1^2+{[6x-x^2]}_2^{3}2\text{units}$