Integral of a particular vector field on manifold

Question

This question is a better reformulation of a part of this question. Suppose $M$ is a Riemannian manifold and let $y \in M$ be a fixed point. Let $$g: M \to T_yM$$be a smooth function. Does it make sense to compute $$ \int_M g(x) \text{ dm}(x) $$? Here m is the Riemannian volume measure. Maybe I can define it using coordinates near $y$, say $\{ \partial_1, \dots, \partial_n \}$, so that $g(x) = g^i(x) \partial_i$ and $$ \int_M g(x) \text{ dm}(x) = \int_M g^i(x) \text{ dm}(x) \partial_i$$ Is this object independent from the choice of the coordinates?