Answer
$$A= \frac{22}{15}$$
Work Step by Step
We will integrate over the $y$-axis; one of the limits will be $y=0$, and the other is the intersection of the two curves. $$y^{2}=2-x$$ $$x=2-y^{2}$$ $$y^{4} = 2-y^{2}$$ $$0 = y^{4} + y^{2} -2$$ $$=(y^{2}+2)(y^{2}-1)$$ $$=(y^{2}+2)(y+1)(y-1)$$ $$y=\pm 1$$ $$x=(\pm 1)^{4}=1$$
$y=\sqrt {2-x}$ (which is $x=2-y^{2}$) is on top.
$$A = \displaystyle\int_{0}^{1} (2-y^{2} - y^{4})dy$$ $$=\left[2y-\frac{1}{3}y^{3}-\frac{1}{5} \right]^{1}_0$$ $$2-\frac{1}{3} - \frac{1}{5} -0$$ $$=\frac{30-5-3}{15}$$ $$= \frac{22}{15}$$
