A scalar field can be accumulated along a curve rather than across an interval or region. If a curve $C$ is parametrized by

$$\mathbf r(t),\qquad a\le t\le b,$$

then its differential arc length is

$$ds=|\mathbf r'(t)|,dt.$$

The line integral of a scalar field $f$ along $C$ is

$$\int_C f,ds

\int_a^b f(\mathbf r(t))|\mathbf r'(t)|,dt.$$

Arc length

Taking $f=1$ gives the length of the curve:

$$L(C)=\int_C1,ds

\int_a^b|\mathbf r'(t)|,dt.$$

For the line segment

$$\mathbf r(t)=(t,2t),\qquad0\le t\le1,$$

we have

$$\mathbf r'(t)=(1,2),$$

so

$$L=\int_0^1\sqrt5,dt=\sqrt5.$$

Accumulating a density

If $f$ is a linear mass density along a wire shaped like $C$, then

$$m=\int_C f,ds$$

is the total mass. The factor $ds$ accounts for actual distance along the curve rather than change in the parameter.

Independence from orientation

Reversing the direction in which the same curve is parametrized does not change a scalar line integral with respect to $ds$. The geometric path and scalar values matter, but its orientation does not.

Vector-field line integrals use a different differential, $d\mathbf r$, and do depend on orientation.