Why can we say here that $\Delta x_i=dx$ as $i$ approaches infinity?

Question

In the proof of the arc length formula we assume that an element of the arc length is $$\Delta L_i=\sqrt{(\Delta x_i)^2+(\Delta y_i)^2}=\sqrt{1+\left(\frac{\Delta y_i}{\Delta x_i}\right)^2}\space \Delta x_i.$$

Applying the Mean Value Theorem

$$\Delta L_i=\sqrt{1+[f^{\prime}(x_i)]^2}\space \Delta x_i.$$

Then to find the total arc length $$L=\displaystyle \lim_{n \to \infty}\sum_{i=1}^{n}\sqrt{1+[f^{\prime}(x_i)]^2}\space \Delta x_i.$$

Now here it is stated that this will be equal to $$\int^a_b \sqrt{1+[f^{\prime}(x)]^2}\space dx. $$ Why did we assume that as $n$ approaches $\infty$ the value of $\Delta x_i= dx$?

Answer 1

$n$ is the number of parts in which you divide the total path length into.

If the total path is a general curve, the approximation (division into $n$ line segments and summing them) will approach the actual answer if the length of each line segment is as small as possible. Thus in order to maintain a constant length the number of parts $(n)$ should approach $\infty$.

As $n$ approaches $\infty$, the path lengths become infinitesimal, thereby allowing us to assume a continuous distribution and replace the sum by an integral and $\Delta x_i$ by $dx$.

Also, as @FShrike mentioned in this comment, writing $\Delta x_i =dx$ is erroneous. Instead, one should think of $dx$ as the a symbol denoting that the variable $x$ is being incremented infinitesimally and summed over a range.