Chapter 4.4, Problem 45E

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 ${\displaystyle {\int }_0^2\left(2y-y^2\right)}dy$. (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.