Let $A, B :[a,b]\subset\mathbb{R}\to\mathbb{R}^2$ be two functions of class $C^{1}([a,b])$ such that two segments (or intervals) $[A(t_1),B(t_1)]$ and $[A(t_2), B(t_2)]$ never intersect for $t_1, t_2\in [a,b], \ t_1\neq t_2$. Prove that the area of the region described by the segment $[AB]$, $R=\{[A(t),B(t)] \ |\ t\in[a,b]\}$, can be computed by the formula:
$$\text{Area}(R)=\int_{a}^b \displaystyle\Big |\det\begin{pmatrix}x(t) & y(t) \\ x'(t) & y'(t) \end{pmatrix}\Big |\ \text{d}t$$
where $B(t)-A(t)=(x(t),y(t)), \ t\in[a,b]$.
P.S. For $U,V\in\mathbb{R}^2$, by the segment $[U,V]$ I mean the set $\{\lambda U+(1-\lambda)V,\ \lambda\in [0,1]\}$.
Question: Can there be found a similar formula in the space? (when $A, B :[a,b]\subset\mathbb{R}\to\mathbb{R}^3)$
I found this propetry in a book of Giuseppe Peano, Applicazioni geometriche del calcolo infinitesimale, 1887, page 238, but I see that his proof is not complete, and I found some really hard problems in trying to fullfil it. You can find the book here: https://archive.org/details/applicazionigeo01peangoog
Note that this property is a nice generalization of Mamikon Theorem (see the book New Horizons in Geometry, by Tom M. Apostol or simply search on the internet), found long time ago by Peano. This is why I'm asking for a generalization in $\mathbb{R}^3$, because Mamikon theorem works for space curves too. For details, see that article, pages 27-28: http://dolecki.perso.math.cnrs.fr/Peano=vulgarization_1307.pdf