Let $g(x)=\dfrac{\int_0^xs^2ds}{\int_0^ns^2ds}$, and let $f(x)=\dfrac{\int_x^ns^2ds}{\int_0^ns^2ds}$. Note that $g(x)+f(x)=1$. Let $v=\{x:f(x)=g(x)\}$, which is just going to be a number and only depends on the numerator since $f(x),g(x)$ have exactly the same denominator. Then I am interested in the quantity $\alpha(n)=\dfrac{v}{n}$.
One can explicitly calculate $v$ (I use $x$ as variable) in terms of $n$. That yields the equation:
$$2x^3-6nx^2+6n^2x=n^3$$
which after plotting it here, we can see it's just a line, namely $v=\beta n$, then $\alpha(n)=\beta$.
That can be confirmed here in this animation. Furthermore, after playing with the animation, if instead of $\int x^2dx$ I consider $\int x^3dx$, the result still holds, namely $\alpha(n)=K$ is just some constant.
This defies my intuition, because I first I assumed the bigger the $n$, the bigger the $\alpha$, in other words, I thought $v$ grew than faster than $\beta n$, but apparently it doesn't.
My questions are:
- I think my analysis is correct, but I'd appreciate if someone can point out any mistake.
- Is there a more general result that deals with this? some theorem that says this is true for all polynomials or something like that?
- My intuition failed me, but is there another way where one can see this without going through all the analysis? in other words, a perspective that makes this fact more straightforward to see?
Edit: The equation I had was wrong. The actual equation is $2x^3=n^3$, as pointed out by @Andrei.