Different expressions for distance & displacement : $\int$$d$$|\vec r|$, $\int$$|$$d$$\vec r$|, and $|$$\int$$d$$\vec r|$ I came across these expressions in my book. And the book says that all these are different from each other. 
The expressions are : $\int$$d$$|\vec r|$, $\int$$|$$d$$\vec r$|, and $|$$\int$$d$$\vec r|$ 
Are they different from each other? 
I know that 


*

*$|$$\int$$d$$\vec r|$ means magnitude of displacement, 

*$\int$$|$$d$$\vec r$| means total distance, 

*But what about $\int$$d$$|\vec r|$? 
I think it should mean total distance too, but I’m not sure if $\int$$d$$|\vec r|$ and $\int$$|$$d$$\vec r$| have the same meaning, do they? 
Edit : Some of the answers say that the question is not very clear, and that a little more explanation would help. I’m not sure what else to add, so I’m attaching a picture of that page
 A: 

In the above Figure-01 we see the quantities $\;\mathrm d\mathbf{r}, \Vert\mathrm d\mathbf{r}\Vert,\mathrm d\Vert\mathbf{r}\Vert\;$ for an infinitesimal displacement from point $\;\boldsymbol{1}\;$ to point $\;\boldsymbol{2}$.
Note that 


*

*$\mathrm d\mathbf{r}\;$ is an infinitesimal vector

*$\Vert\mathrm d\mathbf{r}\Vert\;$  is an always non-negative infinitesimal scalar representing  the magnitude of $\;\mathrm d\mathbf{r}$, and 

*$\mathrm d\Vert\mathbf{r}\Vert\;$  is a real infinitesimal scalar representing  the change of the magnitude of the position vector of point $\;\boldsymbol{1}\;$ to point $\;\boldsymbol{2}$. So, it's positive if increasing, negative if decreasing and zero if constant.


Let see now the integrals of these quantities along a curve $\;C\;$ as in Figure-02.
 
\begin{equation}
\Delta \mathbf{r} \boldsymbol{=}\int\limits_{1}^{2}\!\!\!\!\!\!C\mathrm d \mathbf{r}\boldsymbol{=} \mathbf{r}_2\boldsymbol{-
} \mathbf{r}_1
\tag{01}\label{01}
\end{equation}
\begin{equation}
\Delta\Vert \mathbf{r}\Vert \boldsymbol{=}\int\limits_{1}^{2}\!\!\!\!\!\! C\mathrm d \Vert \mathbf{r}\Vert \boldsymbol{=} \Vert \mathbf{r}_2\Vert\boldsymbol{-} \Vert \mathbf{r}_1\Vert
\tag{02}\label{02}
\end{equation}
So, these quantities depend on the start and end points $\;\boldsymbol{1},\boldsymbol{2}\;$ and are independent of the curve joining them. I don't think that these quantities are useful in any case.
Now,
\begin{equation}
s_{1}^{2} \boldsymbol{=}\int\limits_{1}^{2}\!\!\!\!\!\! C\Vert \mathrm d \mathbf{r}\Vert \boldsymbol{=}\text{lenght of curve between points }\boldsymbol{1},\boldsymbol{2}
\tag{03}\label{03}
\end{equation}

Note :
This textbook is confusing. On one hand it's written that we have a vector 
\begin{equation}
\vec s \boldsymbol{=}\Delta \vec r \boldsymbol{=}\lim_{\Delta \vec r_{i}\boldsymbol{\rightarrow} 0} \sum\limits_{\boldsymbol{i=1}}^{\boldsymbol{i=n}}\Delta \vec r_{i}\boldsymbol{=} \int d\vec r
\tag{A}\label{A}
\end{equation}
and then this same integral of the rhs is used for a scalar, the length of the line joining points 1,2 
\begin{equation}
\int d\vec r\boldsymbol{=}\text{length of the line joining point 1 to 2}\boldsymbol{=}\vert \Delta \vec r\vert
\tag{B}\label{B}
\end{equation}
A: Looks like an integral over radial distance.
A: It is the radial part of the total distance along some path. For example, for a circle there is an angular part of the total distance ($2\pi r$), but there is no any radial part contribution: $dr=0$.
A: Notation matters. You have probably seen  $\int d|\vec{r}|$ written as $\int dr$, without vectors. 


*

*In 1D, this is the same as $\int dx$. The letter for the disttance is not relevant.

*In more than 1 dimension, it's the integral along the radius. For example, in a circle, you'd integrate for all angles and also for all radii from 0 to $R$, for example. There you'd find that integral. $S=\int_0^{2\pi}d\varphi \int_0^R dr$
