The arc length formula. If f' is continuous on [a,b], then the
length of the curve y=f(x), 
, is
The arc length function. If smooth curve C has an equation
y=f(x), , let s(x) be the distance along C from the initial
point 
to the point Q(x,f(x)). Then
(note the change in the dummy variable of integration).
None.
Notice how the formulas change if integrating along the y axis.
Archimedes got the whole ball started by computing the value of 
by
approximating a circle by a regular polygon, both inside and out. He thus
squeezed the value of down to between 
.
He thus approximated a smooth curve by a bunch of line segments, a process
which we continue here. As the line segments get finer and finer, the
approximation gets better and better, until, in the limit, the approximation
becomes exact. We can imagine that we're using an approximation
to f, and hoping that the approximation of the arc length of gets
better and better as the line segments become shorter and shorter.
We do invoke the Mean Value Theorem at some point. You might want to remind yourself of how that works!