Given a compact Riemannian manifold $(M,g)$ and a vector field $X \in \mathfrak{X}(M)$, is it possible to define the integral of $X$ on $M$?
What if $M$ is a Euclidean space? Clearly the definition by components works if we restrict ourselves to affine coordinates, but in general?
Thanks for the help.
I'm not sure whether it is possible or not, but it would require some work, and then one would need to say what it is supposed to mean. In Euclidean space you can do that, because there is a natural way to identify the tangent space of $\mathbb{R}^n$ with $\mathbb{R}^n$ itself, and then you can add vectors from different tangent spaces (integration always has some flavour of adding things and taking the average). On a general manifold there are definitions of parallel displacement of vectors, but the are usually path dependent.
A more common object suitable for integration are differential $(k)$-forms on $(k)$-dimensional manifolds, which are kind of 'dual' to vectors, or more precisely, $(k)$-vectors. You can think of a $(k)$-vector as something representing $k$-dimensional planes and $k$-forms as corresponding $k$-dimensional volume element.
(One of the things you need to keep in mind is that manifolds are usually defined using local charts and glueing them together. Any definition of such an integral has to be independent of this representation, which amount to a certain formula for the transformation of the definition on the intersection of coordinate patches).
One possible exception is the one-dimensional case, where you could define a line integral $\int_c X d\vec{s}$ as $\int_\gamma \langle X,\gamma^\prime\rangle dt$ for any parametrization $\gamma$ of a curve $c$, but this makes implicit use of the fact that a curve is one-dimensional.