Say we are given a space curve described by some parametrization $f(t)=(f_1(t),f_2(t),f_3(t))$. Is there a formula that computes the "area under the curve"?
For example, if $f(t)=(\cos(t), \sin(t),t)$, $0 \leq t \leq 2\pi$, the area under the curve should be half the surface area of the ("uncapped") unit cylinder about the $z$-axis: $2\pi^2$. It turns out this is what I get if compute $\| \int_0^2 f(t)dt \|$, which was what my naive guess at such a formula would be. But this would always return a non-negative number; what if I want an unsigned area?
Additionally, such an integral would have to take into account some form of "orientation," right? I.e., that I'm looking at the "area" between my curve and the $xy$-axis in $\mathbb{R}^3$, as opposed to some other plane. Should I be looking at one of the kinds of integrals that show up in vector calculus (line/path/etc. integrals)?
Thanks in advance for any help!