Question: $$y=x^{\dfrac23}$$ a) Prove that Length of the arc can not be found using the formula $L=\int_{a}^b\sqrt{1+(\frac{dy}{dx})^2}dx$ (call this Formula $F1$) for $x=-1$ to $x=8$.
b) And hence find the Length of arc for the given interval.
My Attempt:
I thought it is fairly easy to prove part (a.) and hence use $L=\int_{a}^b\sqrt{1+(\frac{dx}{dy})^2}dy$ (call this Formula $F2$) to find the answer to (b.).
Hence, I proceeded by showing that $y'=\frac23x^\frac{-1}3$ and since $y'$ is not defined for $x=0$ thus, I concluded (in the back of my mind) that the integral $F1$ shouldn't be defined for the given bounds. But then when I calculated the integrals, both $F2$ (Twice from $y=0$ to $y=1$ and once from $y=1$ to $y=4$) and $F1$ (From $x=-1$ to $x=8$) were defined.
But $F1$ also gives the answer written in the book($=\frac{13\sqrt13+80\sqrt10-16}{27}$). What am I doing wrong? Or is it the question itself?
The same question as this is posted here but nothing had been said regarding this matter.
Edit: $F2$ is giving the correct result after changing the bounds and evaluating as shown in comments by @Shubham.