Area bounded between the parabola y=x^2-2 and the line y=x in square…

The correct answer is Option (1) → $\frac{9}{2}$

$y=x^2-2$

$y=x$

so $x=x^2-2$

so $x=-1,2$

$y=-1,2$

shifting indices to make calculations simplar

let $X=x,Y=y+2$

so new eq.

$Y-2=X^2-2$

$Y=X^2$   ...(1)

for line $Y-2=X$

or $Y=X+2$  ...(2)

area required = $\int\limits_{-1}^2X+2-X^2dx$

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

$=2+4-\frac{8}{3}-\frac{1}{2}+2-\frac{1}{3}=8-\frac{1}{2}-3=5-\frac{1}{2}$

$=\frac{9}{2}$ sq. units