Integral with differential element in a function like $\int |dt|$

Question

so $\Gamma$ is a path of a mass. My book now says: The distance the mass traveled is:

$\int_\Gamma | d\textbf{r} | = \int_\Gamma \sqrt{(dx)^2 + (dy)^2 + (dz)^2}$

Can someone explain me how to interpret $|d\textbf{r}|$ and also why I can write the last integral like that? I'm not used to have the integration variable inside of a function.

Answer 1

This is not meant to be taken verbatim. It is a supposedly "intuitive" typographical picture that expands into $$L(\Gamma)=\int_\Gamma |d{\bf r}|:=\int_a^b\bigl|\dot{\bf r}(t)\bigr|\>dt=\int_a^b\sqrt{\dot x^2(t)+\dot y^2(t)+\dot z^2(t)}\>dt\ .$$

Answer 2

$\lvert d\textbf{r} \rvert$ denotes the norm of the derivative of the vector $\textbf{r}$

This notation (the whole integral) means you are computing a line integral.

So, if $\Gamma$ is the path, and the norm (euclidean norm) can be interpreted as distance, the line integral means you are computing the traveled trajectory.