I'm in doubt in how to precisely define the integral of a vector valued function. And here, I'm not saying: integral of the dot product of two vectors, but integral of the vector itself. For instance, let $\mathbf{v} : A \subset \mathbb{R}\to \mathbb{R}^3$, then I've already seen the integral:
$$\int_A \mathbf{v}\operatorname{d}t$$
Also, I've seem cases like $\mathbf{w} : B \subset \mathbb{R}^3 \to \mathbb{R}^3$ written as:
$$\int_B \mathbf{w} \operatorname{dvol}$$
And in both cases it's integrated in every component. I became very in doubt, because the definition of integral for real-valued functions normally uses upper and lower sums and comparison between then. The point is, those integrals gives vectors back so I have no order relation to establish if the upper sum is greater than the lower sum and etc.
My only try was to define this integral as the limit of a linear combination. For instance, for the first case, we would get:
$$\int_A\mathbf{v}\operatorname{d}t=\lim_{\max\left\{h_i\right\}\to0}{\sum_{i=0}^n}{\mathbf{v}(t_i)h_i}$$
However I don't even know how to state this for a vector valued function $\mathbf{f}:\mathbb{R}^n\to\mathbb{R}^n$ and I don't know how to properly interpret, because vectors are derivations, so what's the meaning of integrating a derivation ? Is it just make a linear combination of a huge number of them ?
Thanks in advance for your help.
This is made precise in Marsden's Vector Calculus. http://www.amazon.com/Vector-Calculus-Jerrold-Marsden/dp/1429215089