Is there how to measure the distance between two points?

Question

The function $$ y = x^3 - x^2 + x $$

Is there how to measure the distance between two points? Not the shortest, but the real size of the line between these points.

Answer 1

Yes, there is.

If $f$ is a differentable function defined at $[a,b]$ and this integral exists and it is finite $$\int_a^b\sqrt{1+f'(x)^2}dx$$ then it is the length of the curve $y=f(x)$ between $(a,f(a))$ and $(b,f(b))$.