Question

Which of the following definite integrals could be used to calculate the total area bounded by the graph of y = x^3 – 3x^2 + 2x and the x-axis?

Answer 1

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

$\int_0^2f(x)\,\mathrm dx=2\int_0^1f(x)\,\mathrm dx$