Intuition Wanted: Why Define Integrals Component-Wise
Addition, subtraction, scalar multiplication, scalar division, limits, and differentiation all act component-wise on vectors. You would need a good reason to make integration inconsistent with those!
For the "deeper level": However we define integration of vectors, we want it to be:
- Linear, $\int(a+b)=\int a+\int b$
- Covariant with vector space isomorphisms, $\int T f=T\int f$.
- Consistent with the usual embedding of $\mathbb R$ in $\mathbb R^n$, $\int (f, 0, 0, \ldots)=(\int f, 0, 0, \ldots)$.
These axioms force it to act component-wise.
The Riemann Integral, say, is based on sums. The sums of vectors are defined component-wise. And different norms on $\mathbb R$ are topologically equivalent. Therefore, this is the exact thing you would end up with anyway for Riemann integrals if you defined them by analogy instead of component-wise explicitly.
On a much more general level, considering sets, mapping into a product $f:A\to B\times C$ is essentially really just two mapping $f_B:A\to B$ and $f_C:A\to C$. Similarly, mapping out of a disjoint union of sets is just two functions. This extends to products of more than two things and in particular to functions to $\mathbb R^n$. This phenomenon justifies the reduction you mention. It is a categorical observation valid in many different contexts.