Answer
$\dfrac{9}{2}$
Work Step by Step
Let us consider that two continuous functions $f(x)$ and $g(x)$ with $f(x)\geq g(x)$ on the interval $[a,b]$ . The area (A) of the region bounded by the graph of $f(x)$ and $g(x)$ on the interval $[a,b]$ can be calculated as: $Area(A)=\int_a^b [f(x)-g(x)] \ dx$
The given curves are $y=x$ and $y=x^2-2$.The intersection points of these curves satisfy the given equations are: $x=x^2-2 \implies x^2-x-2=0 \\ (x+1)(x-2)=0 \\ x=-1 \ \text{and}\ x=2$ . These points will be the lower and upper limit of the integration.
Thus, the area of the region is:
$A=\int_a^b [f(x)-g(x)] \ dx= \int_{-1}^2 [x-(x^2-2)] \ dx\\ =[\dfrac{x^2}{2}-\dfrac{x^3}{3}+2x]_{-1}^2\\=\dfrac{9}{2}$
