jmc

The graph of $x=f(y)$ can be drawn by reflecting the graph of $y=f(x)$ across the line $y=x$: The enclosed region, shown above in gray, is bounded by two quarter-circle arcs. The portion above and to the left of the red dashed line has area $\frac{\pi }{4}-\frac{1}{2}$, since it is a quarter of a unit disc minus a right triangle of base and height $1$. The portion below and to the right of the red dashed line is the same. Thus, the total enclosed region has area $\frac{\pi }{2}-1$; rounded to the nearest hundredth, this is $\boxed{0.57}$.