I would like to find the angle subtended by an arc of a circle with a changing radius. The main issue is that the radius is changing by a non-linear factor as shown below:
The integral on the left is correct for f(s)=0. It will just give the arc length formula which gives theta.
f(s) is changing wrt the coordinate s. Do I need to find a differential form of f(s) as I did with the arc length formula? i.e. Going from s/R to ds/R for the integration.
I guess #2 is where my real confusion stems from. I need to find the differential form of s/R to properly integrate and find the angle subtended. I can't just add a "ds" and integrate then I would just get:
So, then if i needed to find the differential form of s/R, then wouldn't I need to find some sort of differential form of f(s)? Or can I just plug f(s) in and integrate?