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.
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what two points? $y=x^3 -x^2 + x$ is a collection of infinite points – Dando18 Jun 02 '17 at 16:23
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what kind of points? – Dr. Sonnhard Graubner Jun 02 '17 at 16:23
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coordinates (x,y) – user2793412 Jun 02 '17 at 16:24
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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 $(b,f(b))$.
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