Raining in Flatland

Drawing the graph: $F(x)=x(x^2-5x+7)$ Since discriminant of $x^2-5x+7$ is negative therefore graph will intersect x axis only at origin. $F'(x)=3x^2-10x+7=(3x-7)(x-1)$ $\implies$ graph is increasing in $(-\infty,1)$ and $(\dfrac 73,\infty)$ while its decreasing in $(1,\dfrac 73)$ hence attaining a local maxima at $x=1$ and a local minima at $x=\dfrac 73$ along with having horizontal tangents at $x=1$ and $x=\dfrac 73$ . Hence the graph would look like. Clearly required region is : $\int_1^3\left(3-(x^3-5x^2+7x)\right)\, dx$ $=\Large \dfrac 43\approx \boxed{1.333}$