It is possible to derive the arclength analytically by using the Pythagorean theorem:
given a curve y(x), infinitestimal length dl along the curve can be given as: $(dl)^2 = (dx)^2 + (dy)^2$
Dividing both sides by $(dx)^2$, $(dl/dx)^2 = 1 + (y')^2$
Thus, $$l = \int \sqrt{1 + (y')^2}\,dx$$
For a surface defined parametrically, $z(x,y)$, the infinitestimal area $ds$ should be: $(ds)^2 = (dx\, dy)^2 + (dx \,dz)^2 + (dz\, dy)^2$
How do I actually arrive at this (the aforementioned equation)? I tried multiplying $dz$ by $dl$ to get the area of an infitestimal parallelogram, but that doesn't work. I know I can do it using vectors, but what is the analytic approach?