In 2D one may use Green's theorem to calculate the area of a closed curve quite easily.
Assume the curve is parametrized by:
$r(t)=\langle x(t),y(t)\rangle$
$a \leq t \leq b.$
Then we get:
$dx=x'(t)dt$; $dy=y'(t)dt.$
And finally through a clever application of Green's theorem:
$$A = \frac{1}{2}\int^b_a \big(-y(t)x'(t)+x(t)y'(t)\big)dt$$
Does this mean that Stoke's theorem can be used to calculate the "minimal" area described by a curve in 3D?
If yes how, if not why not?