Understanding the arc length integral formula I believe the proof in my book is slighty more informal than the proof that uses the Mean Value Theorem. Could someone tell me what exactly the difference is, and if there are any mistakes in the proof below? Thanks.

Proof of the arc length integral formula
Divide your interval $[a,b]$ into $n$ pieces of width $\Delta x$, then zoom into the subinterval $[x_{i-1},x_i]$. The arc length in this interval is approximately $$\sqrt{\Delta x^2+\Delta y^2}=\sqrt{1+\left(\frac{\Delta y}{\Delta x}\right)^2}\Delta x$$
As $\Delta x$ goes to zero, $\frac{\Delta y}{\Delta x}$ is equal to the slope at $x=x_{i-1}$, that is $f'(x_{i-1})$.
The Riemann sum becomes $$\sum_{i=1}^n \sqrt{1+[f'(x_{i-1})]^2}\Delta x$$
As $n\to\infty$, the arc length is $$\int_a^b \sqrt{1+[f'(x)]^2}\,dx$$

Elaboration of my comment:
You want to convert an infinite sum to an integral. As you probably understand that can be interpreted as summing infinitely many rectangles and deciding what the area converges to. But this is not enough! The rectangles have to be infinitely small as well. Take a look at the picture. There are infinitely rectangles, but since they are not infinitely small, the area does not converge to the area under the curve.

