# Find the area of the following region using integration :

$\\y=x+2, \text { } x \geq 0 \\ y=-x+2, \text { } x<0 \\$

$x+2= x^2 \\$

$x^2 -x -2 = 0 \\$

$(x-2)(x+1)=0 \\$

$x=2,-1(x=-1 \text { is not possible })$

$-x+2= x^2 \\$

$x^2 +x -2 = 0 \\$

$(x+2)(x-1)=0 \\$

$x=-2,1(x=1 \text { is not possible })$

The Required Area = Area of ABCOA

\begin{aligned} \text { Area of ABCOA } &=2\left[\int_{0}^{2}(\mathrm{x}+2) \mathrm{d} \mathrm{x}-\int_{0}^{2} \mathrm{x}^{2} \mathrm{dx}\right] \\ &=2\left[\frac{(\mathrm{x}+2)^{2}}{2}-\frac{\mathrm{x}^{3}}{3}\right]_{0}^{2} \\ &=2\left[6-\frac{8}{3}\right] \\ & = \frac{20}{3} \mathrm{square} \text { units } \end{aligned}

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