how $\int_a ^ b |f'(x)|$ gives the length of the arc of the contour $f$ : $(f(x) : x \in [a , b])$

Question

I got to know that $\int_a ^ b |f'(x)|$ gives the length of any contour. Where $f(x)$ is a piece-wise differentiable function from $[a,b]$ to $\mathbb R^2$. I was reading complex integral . Can anyone please enlighten me how it is ? I understand for the functions whose Domain and Range are $\mathbb R$.

Answer 1

What is the length of a curve?

Let $a = x_0 < x_1 < x_2 < ... < x_n = b$ a partition of $[a,b]$

An approximation of the length can be

$Len(f) \approx \sum_{k=0}^{n-1}|f(x_i+1)-f(x_i)|$

You can see that when at the limit where the partition is delicate enough you get

$Len(f) = \sum_{k=0}^{\infty}|f(x_i+1)-f(x_i)|=\sum_{k=0}^{\infty}\Delta x_i|\frac{f(x_i+1)-f(x_i)}{\Delta x_i}|\rightarrow_{n\rightarrow \infty} = \int_a^b|f'(x)|dx$

Thats the general idea