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