Dummy variable?

[KindofSlow]

I’m working on Arc Length with Vector Functions where:
Arc length is L = ∫ from a to b of magnitude of r’(t) dt
Then arc length function is s(t) = ∫ from 0 to t of magnitude of r’(u) du

I'm trying to get all the functions and variables and the relationships between them straight in my head.

So my really simple question is – “Is u in the arc length function a “dummy variable”” ?

Hope this makes sense.
Thank you

 

[KindofSlow]

I figured out the relationships.
I don't know if u is a "dummy variable" or not but I don't think it matters.
No need to reply.
Thank you

 

[HallsofIvy]

Reply anyway! Yes, u here is a dummy variable because, once you have done the integration, the result is a function of t only. For example, it I wanted to find t so that is distance, along the curve, from the point where t= 0, is 1, I could do it as \(\displaystyle \int_0^t |r'(u)| du\) or \(\displaystyle \int_0^t|r'(v)|dv\) or \(\displaystyle \int_0^t [r'(w)]dw\), etc. The choice of "variable of integration" is irrelevant to the result.

(Strictly speaking "dummy" means simply "unable to speak" as in "dumb animal". It's more general use is something a slur on deaf people!)