Let [tex]f(x)=x^3-3x^2+2x=x(x-1)(x-2)[/tex]. Suppose we define a new function that is [tex]f[/tex] transformed so that it is symmetric about the origin. This can be done with
[tex]g(x)=f(x+1)=(x+1)(x)(x-1)[/tex]
Notice that [tex]g[/tex] is an odd function, since
[tex]g(-x)=(-x+1)(-x)(-x-1)=-(x-1)(x)(x+1)[/tex]
So if [tex]g(x)[/tex] is symmetric about the origin, this means [tex]f(x)[/tex] must by symmetric about the point [tex](1,0)[/tex]. This in turn means that the area between [tex]f[/tex] and the x-axis is the same from 0 to 1 as it is from 1 to 2. This means
[tex]\int_0^2f(x)\,\mathrm dx=2\int_0^1f(x)\,\mathrm dx[/tex]
