Answer
$\displaystyle \frac{9}{2}$
Work Step by Step
Graphing the given equations, we find the intersections
or we solve the equation $f(y)=g(y), $ where
$ f(y)=y+2 \quad$ (the curve on the right side) and
$g(y)=y^{2} \quad$ (the curve on the left).
When $y\in[c,d]=[-1,2]$, the area between the graphs is
$A=\displaystyle \int_{c}^{d} [f(y)-g(y)]dy=\displaystyle \int_{-1}^{2}[y+2-y^{2}]dy=$
$=\displaystyle \left[\frac{y^{2}}{2}+2y-\frac{y^{3}}{3}\right]_{-1}^{2}$
=$ \displaystyle \frac{(4-1)}{2}+2(2+1)-\frac{(8+1)}{3}$
$=\displaystyle \frac{9}{2}$
