Derivation of integral version of arc length formula

Question

I apologize in advance if this question is too vague, and I am happy to try to clarify anything. I'd like to show that one can go from the definition of the length of a curve (as the least upper bound of the set of all numbers $|\pi(P)|$, where $|\pi(P)|$ is an inscribed polygon whose vertices correspond to the chosen partition P), to the formula $$ \int_a^b \sqrt{1 + f'(x)} \, dx $$ without using a parametrization of the curve and without referencing Riemann sums. I'm following with Apostol's calculus book and he shows this via using parametric equations (he shows that the arc length is equal to the integral of the speed then treats the graph of a real-valued function as the vector-valued function $r(t) = \langle t, f(t) \rangle$ and calculates the speed). Is it possible to do this without this approach and also without the standard "using the Mean Value Theorem....we get a Riemann sum....which becomes the integral...?" Maybe using upper and lower sums and then invoking the definition of the integral (the only value that lies between all upper and lower sums for all partitions)?

Answer 1

While learning calculus I didn't know if there exists any formula to find the length of arc but I always believed that we can find the length of arc using integration this is what I had done.

• Pythagoras Formula

$$\begin{align*} \delta s & = \sqrt{(\delta x)^2 + (\delta y)^2}\\ & \frac {\delta s}{\delta x}= \sqrt {(1)^2+\left(\frac {\delta y}{\delta x}\right)^2}\\ \end{align*}$$ The rest you can use integration for the same.

though in vector calculus we'd used tangent properties to find the length of arc which you can find in the book Engineering Mathematics by Daniel G Zill