If you take a look at the derivation section of line integral article, you can see the following statement:
$$I = \lim_{\Delta s_i \rightarrow 0} \sum_{i=1}^n f(\mathbf{r}(t_i)) \, \Delta s_i. \tag{1}$$ We note that, by the mean value theorem, the distance between subsequent points on the curve, is
$$\Delta s_i = |\mathbf{r}(t_i+\Delta t)-\mathbf{r} (t_i)| \approx |\mathbf{r}' (t_i)| \,\Delta t.$$
Substituting this in the above Riemann sum yields
$$I = \lim_{\Delta t \rightarrow 0} \sum_{i=1}^n f(\mathbf{r}(t_i))|\mathbf{r}'(t_i)| \, \Delta t \tag{2}$$
I'm wondering how to more rigorously derive eq. 2 from eq. 1, because I have a few doubts about the method used above: Note that it is a vector valued function. The mean value theorem article that it refers to doesn't really give any explanation, it even states there is no direct analog of mean value theorem for vector valued function (actually there's one involving an inequality).
There is no exact analog of the mean value theorem for vector-valued functions.
I will be grateful for a clear explanation as I'm not a math expert.