## Multivariable integration

In this post I’ll consider the task of integrating a function $f : \mathbb{R}^n \to \mathbb{R}$ operating under the assumption that it is bounded. We will generalize the notion of a closed interval to that of a closed cell, and then generalize the notion of a partition to that of a division. We should note that in one variable, a partition can alternatively be viewed as a set of closed intervals on the real line, such that

1. The union of all the closed intervals is the whole interval in question, $[a,b]$.
2. The interiors of these closed intervals have empty intersection, that is, they are disjoint.

It should come as no surprise, then, that in the multivariable case, we have the following definition.

Definition. A closed cell in $\mathbb{R}^n$ is a set of the form $[a_1, b_1] \times [a_2, b_2] \times \ldots \times [a_n, b_n]$. That is, a closed cell in $\mathbb{R}^n$ is the Cartesian product of $n$ closed intervals in the real line.

To provide some intuition, note that a 1-cell is precisely a closed interval, a 2-cell is a closed rectangle, a 3-cell is a closed cube, and so on. We now proceed with our generalization of a partition.

Definition. Let $P$ be a closed cell in $\mathbb{R}^n$. A division of $P$ is a set $\Delta = \{ P_1, P_2, \ldots, P_k \}$ where

1. Each $P_i$ is a closed cell
2. $\bigcup_{i=1}^k P_i = P$
3. $P_i \cap P_j = \emptyset$ for $i \neq j$

Recall that in the single-variable case, the upper and lower Riemann sums were formed by taking the sum of the areas of rectangles of a certain height. For the upper sum, the height of these rectangles was the supremum of the function $f$ over the subinterval in question. For the lower sum, the height was given by the infimum of the function. Note how we started with closed intervals in the function’s domain and constructed rectangles. That is, we started with 1-cells in the domain, and we constructed 2-cells, based on the function’s behaviour.

In the multivariable case we will start with n-cells in our domain $P$ and use the function’s supremum and infimum over each cell to construct an n+1-cell. We now define the volume of a cell, which is quite a simple concept: it is merely the product of the cell’s dimensions. So the volume of a 1-cell is length, the volume of a 2-cell is area, and so forth.

Definition. Let $P = [a_1,b_1] \times \ldots \times [a_n,b_n]$ be a closed cell in $\mathbb{R}^n$. Then its volume is given by $\mathrm{vol}(P) = (b_1 - a_1) \cdots (b_n - a_n)$.

We are finally ready to define what are known as the upper and lower Darboux sums for the function $f$ on the closed cell $P \subseteq \mathbb{R}^n$.

Definition. Let $f$ be a bounded function defined on a closed cell $P \subseteq \mathbb{R}^n$. Let $\Delta = \{P_1, P_2, \ldots, P_k\}$ be a division of $P$. Then the upper and lower Darboux sums for $f$ with respect to $\Delta$ are given, respectively, by

$\mathrm{U}(f,\Delta) = \sum\limits_{i=1}^k \mathrm{vol}(P_i) \cdot \sup\limits_{P_i}(f)$
$\mathrm{L}(f,\Delta) = \sum\limits_{i=1}^k \mathrm{vol}(P_i) \cdot \inf\limits_{P_i}(f)$

where we use the shorthand notation,

$\sup\limits_{P_i}(f) = \sup\{ f(x) \mid x \in P_i \}$
$\inf\limits_{P_i}(f) = \inf\{ f(x) \mid x \in P_i \}$

Let us consider some closed cell $P \subseteq \mathbb{R}^n$. Note that the upper and lower sums associated to a function $f$ can certainly change if we use a different division of $P$ (that is, we “chop it up” differently.) So it makes sense to consider the set of all upper and lower sums as we vary the division.

Definition. Let $P$ be a closed cell in $\mathbb{R}^n$, and let $f : P \to \mathbb{R}$ be a function. We define the upper and lower integrals of $f$ over the cell $P$ by

$\overline{\int_P} f = \inf\{ \mathrm{U}(f, \Delta) \mid \Delta \text{ a division of } P\}$
$\underline{\int_P} f = \sup\{ \mathrm{L}(f, \Delta) \mid \Delta \text{ a division of } P\}$

and we say, in the case that $\overline{\int_P} f = \underline{\int_P} f$, that the function $f$ is integrable on the closed cell $P$.

Anyways, there is much more to say on the subject. If your interest is piqued you can see the relevant section in my LaTeXed MATH 247 notes (which by the way aren’t done yet, so no guarantees on whether they’re perfect and polished, heh.)