Line Integral w/ Respect to Arc Length & x/y

[yifli]

There are line integral with respect to arc length and line integral with respect x/y.

I know $\int_C Pdx+Qdy$ is useful to calculate the work. When do we need the line integral with respect to arc length?

 

[LCKurtz]

There are line integral with respect to arc length and line integral with respect x/y.

I know $\int_C Pdx+Qdy$ is useful to calculate the work. When do we need the line integral with respect to arc length?

Suppose you have a wire with a density which varies along its length. You need an arc length type integral to evaluate its mass.

$$M =\int_C \delta(s)\,ds$$