Line integral with respect to arc length

[Castilla]

In a line integral with respect to arc length, we have something like f(x, y)ds "inside" the integral sign.

The ds tells us that we are working with the arc length function s, taking diferences (s_K+1 - s_k) in the sums that tend to the line integral.

Question: do we shall understand that x = x(s), y = y(s), or x and y are functions of another parameter? In this last case, this means that we work with two different parameters in the sums that tend to the integral?

Thanks for your help.

 

[mathman]

Ultimately x and y must be functions of s, although they be expressed in terms of another parameter.
Specifically ds² = dx² + dy².

 

[Castilla]

I think I have understood.

s = s(t), but s is an increasing function, so we have its inverse t = t(s). Then
x = x(t) = x(t(s)), y = y(t) = y(t(s)). Is this ok?

 

[HallsofIvy]

If t is an arbitrary parameter, it does not necessarily follow that s is an increasing function of t.