Your first priority should be to get a good handle on differential forms and on Stokes's theorem for differential forms and manifolds with boundary, say out of Munkres' Analysis on Manifolds, Spivak's Calculus on Manifolds, or Madsen's From Calculus to Cohomology, after which you should be able to look at the more introductory accounts of geometric measure theory; graduate-level introductions will also require some Lebesgue theory. Roughly speaking, what's going on is the following:
- Integrating $k$-forms over a given $k$-dimensional oriented submanifold (possibly with boundary) $\Sigma$ defines a linear functional
$$
[\Sigma] : \omega \mapsto \langle [\Sigma],\omega \rangle := \int_\Sigma \omega
$$
on the vector space of $k$-forms, and hence, by definition, a $k$-current $[\Sigma]$; if you pick the appropriate
topological vector space of suitable (e.g., compactly supported)
$k$-forms, then your submanifold really does define a continuous
linear functional and hence a $k$-current in the strict analytic sense. In particular, a $0$-current (correctly defined) is just a distribution in the sense of analysis.
- If you think of $k$-dimensional oriented submanifolds with boundary as defining functionals on the (appropriate) space of $k$-forms, then Stokes's theorem tells you that taking the operation of taking the boundary of a $(k+1)$-dimensional submanifold (and hence getting a $k$-dimensional submanifold as the boundary) is the transpose of taking the differential of a $k$-form (and hence getting a $(k+1)$-form as the differential):
$$
\langle [\partial M], \omega \rangle = \int_{\partial M} \omega = \int_M d\omega = \langle [M], d\omega \rangle,
$$
so $d^\ast[M] = [\partial M]$, i.e., the differential-transpose $d^\ast[M]$ of the $k$-current $[M]$, which is a $(k+1)$-current, is the $(k+1)$-current $[\partial M]$ defined by the boundary $\partial M$ of $M$.
So, what's actually going on in your example?
- On the one hand, the interval $[a,b]$, oriented from $a$ to $b$ (i.e., left-to-right if $a < b$, and vice versa otherwise), is a $1$-dimensional oriented submanifold with boundary of $\mathbb{R}$, and its boundary is $\partial[a,b] = (\{a\},-) \sqcup (\{b\},+)$, i.e., the $0$-dimensional oriented submanifold $\{a,b\}$ of $\mathbb{R}$ where the connected component $\{a\}$ is negatively oriented and the connected component $\{b\}$ is positively oriented. Hence, the interval $[a,b]$ defines a $1$-current $[[a,b]]$ and its boundary $\partial[a,b] = (\{a\},-) \sqcup (\{b\},+)$ defines a $0$-current $$d^\ast[[a,b]] = [\partial[a,b]] = [(\{a\},-) \sqcup (\{b\},+)];$$ in fact, if we exploit the vector space structure on distributions and take a single point $\{c\}$ to have the positive orientation by default, we can even do one better and write
$$
d^\ast[[a,b]] = [\partial[a,b]] = [\{b\}] - [\{a\}].
$$
- On the other hand, the function $f(x) = x^2$ defines a $0$-form on $\mathbb{R}$ with differential the $1$-form $df = f^\prime(x)\,dx = 2x\,dx$.
So now, given that integrating a $0$-form $g$ over a (positively oriented) point $c$ is simply evaluating $g$ at $c$, i.e.,
$$
\langle [\{c\}], g \rangle := \int_{\{c\}} g := g(c),
$$
whilst integrating a $1$-form $\omega = h(x)\,dx$ over an oriented interval $[s,t]$ is simply integrating $h$ over $[s,t]$, i.e,
$$
\langle [[s,t]], \omega \rangle := \int_{[a,b]} \omega := \int_a^b h(x)\,dx,
$$
you can reinterpret the computation $\int_a^b 2x\,dx = b^2-a^2$ as
\begin{align*}
\int_a^b 2x\,dx &= \int_{[a,b]} df\\ &= \langle [[a,b]],df \rangle\\ &= \langle d^\ast[[a,b]],f \rangle\\ &= \langle [\{b\}]-[\{a\}],f\rangle\\ &= \langle [\{b\}],f\rangle - \langle [\{a\}],f\rangle\\ &= \int_{\{b\}} f - \int_{\{a\}} f\\ &= f(b) - f(a)\\ &= b^2 - a^2.
\end{align*}
It's a lot of notation, but already, one advantage of working in terms of currents is the flexibility to take linear combinations of submanifolds, especially when taking oriented boundaries of oriented submanifolds with boundary—this leads straight to the notion of a de Rham homology of currents in duality with the de Rham cohomology of differential forms. However, I imagine that for a lot of people the real fun begins with connections to Plateau's problem and the theory of minimal surfaces, which leads into geometric measure theory.
