# The area (in sq. units) of the region  $A=\left \{ \left ( x,y \right ):x^{2}\leq y\leq x+2 \right \}$ is : Option 1) $\frac{10}{3}$ Option 2) $\frac{9}{2}$ Option 3)  $\frac{31}{6}$ Option 4)  $\frac{13}{6}$

Point of intersection of $x^{2}=y$ and $y=x+2$

$\left ( -1,1 \right )$ and $\left ( 2,4 \right )$

$y=x+2$

Required Area,

$=\int_{-1}^{2}\left ( x+2-x^{2} \right )dx$

$=\left [ \frac{x^{2}}{2}+2x-\frac{x^{3}}{3} \right ]^{2}_{-1}$

$=\frac{9}{2}$

Option 1)

$\frac{10}{3}$

Option 2)

$\frac{9}{2}$

Option 3)

$\frac{31}{6}$

Option 4)

$\frac{13}{6}$

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