Integral of absolute value of dx

Question

How do you evaluate $\int_1^0\vert \mathrm{d}x\vert$ and what is the reason behind it? I've seen two answers:

1) $\vert 0\vert-\vert1\vert=-1$

2) $\vert0-1\vert=1$

So which one is it?

Answer 1

This notation is incorrect - it is not defined and does not make sense.

Referring to the physics context that you provide, when we perform a line integral $$ \int \vec{E} \cdot d\vec{x} $$

what this notation means is that we are integrating alone a directed path of integration, the dot product of a vector field with the infinitisimal vector pointing in the direction of our path. Note that, as we move along a path of integration, our direction vector may change

In the case of the problem you provided, the electric field is pointing horizontally to the right, so if we choose a straight path parallel to the electric field, oriented in the same direction (moving from left to right), the electric field and the direction vector of our integration path point in the same direction, always (our path doesn't change direction as we go):

Since the electric field and the direction vector always point in the same direction: $$ \vec{E} \cdot d\vec{x} = E \, dx $$ What the left side means is the dot product of electric field with an infinitesimal direction vector and what the right side means is the magnitude of the electric field times an infinitesimal magnitude of length.