Integral of 1-forms and line integrals Let $X \subseteq \mathbb R^n$, $\lambda: [a,b] \to X$, and $\omega :X \to (\mathbb R^n)^*$. The integral of the 1-form $\omega$ along $\lambda$ is defined as:
$$ \int_\lambda \omega = \lim_{|P|\to 0} \sum_{j=1}^k \omega(\lambda(\xi_j))(\lambda(t_j)-\lambda(t_{j-1})) $$
where, $P=\{t_0<t_1<\cdots < t_k\}$ is a tagged partition and $t_{j-1} \leq \xi_j \leq t_j$.
On the other hand, if $f:X \to \mathbb R$, then the line integral of $f$ along $\lambda$ is defined as:
$$ \int_\lambda f \, ds = \lim_{|P|\to 0} \sum_{j=1}^k f(\xi_j) \Vert \lambda(t_j)-\lambda(t_{j-1}) \Vert $$
Is the integral of 1-forms a generalization of line integrals? That is, given $f:X \to \mathbb R$, is there a 1-form $\omega :X \to (\mathbb R^n)^*$ such that
$$ \int_\lambda f \, ds = \int_\lambda \omega$$
holds?
 A: No, you cannot find a $1$-form on $X$ whose restriction to the parametrized curve $\lambda$ is $ds$ (which is notation and not a $1$-form itself) for $\lambda$. (For example, note that $\int_\lambda f\,ds$ does not depend on the orientation of $\lambda$, whereas the integral of any $1$-form does.)
However, if you fix a parametrized curve $\lambda$, you can in some situations find a $1$-form whose restriction to $\lambda$ will give $ds$ for that particular curve. For example, if $X=\Bbb R^2$ and you know the outward pointing unit normal $\vec n = (P,Q)$ to $\lambda$ at points $(x,y)\in\lambda$, then you can easily check that $ds = -Q\,dx + P\,dy$. (There are analogous formulas for integrating over hypersurfaces in $\Bbb R^n$.) So, in this situation, you can compute the integral $\int_\gamma f\,ds$ by integrating the $1$-form $f(-Q\,dx + P\,dy)$. This generalizes to any surface with a Riemannian metric.
REMARK: As a remark way beyond the scope of this question, one of the most tantalizing features of complex differential geometry is the following. Given an $n$-dimensional complex (Kähler) manifold $X$, for every $k$ between $1$ and $n$ there is a differential ($2k$-) form that gives the induced volume element for any arbitrary $k$-dimensional complex submanifold of $X$.)
