The area of the region that lies to the right of the $y$-axis and to the left of the parabola $x=2y-y^2$ (the shaded region in the figure) is given by the integral $\int_{0}^{\,2} {\left(2y-y^2\right)}\,dy.$ (Turn your head clockwise and think of the region as lying below the curve $x=2y-y^2$ from $y = 0$ to $y = 2$.) Find the area of the region.

Area =