TITLE: Intuition Wanted: Why Define Integrals Component-Wise
QUESTION [17 upvotes]: In our analysis course, we just defined the following:

Let $g := (g_1, \ldots, g_n) \colon [a, b] \to \mathbb{R}^n$, where $g_1, \ldots, g_n \colon [a,b] \to \mathbb{R}$ are integrable.
Then we call the integral of $g$ over $[a,b]$
\begin{equation*}
\int_{a}^{b} g(t) \ \text{dt}
:= \begin{pmatrix}
\int_{a}^{b} g_1(t) \ \text{dt} \\
\vdots \\
\int_{a}^{b} g_n(t) \ \text{dt}
\end{pmatrix}.
\end{equation*}

I came across this definition again at the beginning of measure theory, when we stated:

We ultimately want to integration functions $\mathbb{R}^n \to \mathbb{R}^m$, but because of the above definition we can, without loss of generality, restrict ourselves to the case $m = 1$.

My Question
What is the intuition behind this definition, why does it make sense, if you will, ''on a deeper level''?

REPLY [3 votes]: The idea of component-wise integrals has a concrete geometrical meaning: to decompose a function curve by projecting it onto planes, then to take the (signed) areas of these projections in their respective planes.
For example, let's take a look at a function v(t)=(x(t),y(t)). In the picture below, the two components of the integral of v are the area of the red curve in its horizontal plane and the area of the yellow curve in its vertical plane.

Another way to define the integral of v would be to take the integral of the norms ||v(t)||, but I guess component-wise integrals have more respect for structure and thus have nicer algebraic properties.
We can also look at a physical interpretation, to make this even more concrete. In the above, interpret t as time, and v(t) as the velocity of an object O moving within some plane, so that x(t) is its horizontal velocity and y(t) is its vertical velocity, at any instant t. Then the integral of x is the horizontal distance traveled by O, and the integral of y is the vertical distance traveled by O, so that the component-wise integral of v is the 2D distance traveled by O. So with component-wise integrals, the integral of velocity remains the distance, even in multidimensional situations.
(Illustration produced thanks to Octave and this script.)