A misunderstanding I had in calculus

Preamble:
  I learned basic calculus at school, but curiously never got back to it (apart from basic measure theory) at university where this was considered as being part of "applied mathematics". Certainly we never did multivariate calculus at univerisity.

Since this keyboard does not allow me ro write integration symbols and since I am a bit too lazy to draw a picture I'm going to represent it by the word "integrate".

At school we had always studies things like
integrate f(x) dx
Sure we had been give meaning of this but with time I began thinking of this as "find the primitive of f".
I then got the idea that the d in dx was rather pointless, it was just to indicate which of the parameters was the variable.

But when I got to advanced calculus on my own this way of thinking really prevented me from understanding expressions like

integrate f(x).ds
(for a line integral, we've left out the curve parameter)
where the dot above  is supposed to represent the dot product.

Now this time I was pretty stumped for several reasons:
1. Here s is not a variable to be found anywhere in f(x)
2. The dot product with ds really invalidates my previous approach as to the role of ds.

I must point out that the notation using dot ds is not perfect, because s is not a variable name that can be replaced by another one, it has a specific meaning of an infinitessimal tangent vector and this has to be recognised!