Multiple Integrals over Rectangles

Consider a solid tent-shaped region sitting above a rectangular floor in the $xy$ -plane. Its height at each point $(x,y)$ is given by some function $f(x,y)$ . What number should we call its volume? For a single-variable function, we sliced the region under a curve into thin vertical strips. We then summed rectangular areas. The same idea works in higher dimensions. But now we must slice a rectangle in $\mathbb{R}^n$ into a grid of tiny sub-rectangles, rather than an interval into sub-intervals. We must track how the resulting sums behave as the grid gets finer in every direction simultaneously. This construction matters because it is the rigorous foundation beneath every multivariable volume, mass, flux, and probability computation. Physicists, engineers, and statisticians all rely on it. Before one can legitimately swap the order of integration, change variables, or integrate over a disk or a sphere, one needs a precise notion. One must know what "the integral of $f$ over a rectangle $R \subset \mathbb{R}^n$ " even means. One also needs a workable test for when that number exists. The big picture is a direct generalization of the one-variable Riemann integral. First, partition the rectangle into a grid of sub-rectangles. Then form upper and lower sums using the supremum and infimum of $f$ on each piece. Declare $f$ integrable when these sums can be forced arbitrarily close together by refining the grid. Continuity — building on Limits and Continuity in Several Variables — combined with compactness of closed rectangles, will guarantee this happens. This holds true for a large and useful class of functions. This article builds the definition of partitions and Riemann sums over a rectangle $R \subset \mathbb{R}^n$ . It proves the fundamental integrability criterion (Riemann's criterion). It also shows continuous functions on compact rectangles are always integrable. It sets the stage for later results. These include Fubini's Theorem, which reduces multiple integrals to iterated single integrals. It also sets the stage for the extension of integration to non-rectangular regions.

Key takeaways

A rectangle $R = [a_1,b_1]\times\cdots\times[a_n,b_n] \subset \mathbb{R}^n$ is partitioned by partitioning each side, producing a grid of sub-rectangles.
Upper and lower Riemann sums are built from suprema/infima of $f$ on each sub-rectangle, weighted by sub-rectangle volume.
The upper integral is the infimum of upper sums; the lower integral is the supremum of lower sums; $f$ is integrable when they agree.
Riemann's criterion: $f$ is integrable on $R$ iff for every $\varepsilon>0$ there is a partition with upper sum minus lower sum less than $\varepsilon$ .
Every continuous function on a closed, bounded rectangle is integrable — this relies crucially on compactness giving uniform continuity.
Key idea: integrability is a statement about how well suprema and infima on a grid can be squeezed together as the grid refines in every coordinate direction at once.

Where this lives today

The Riemann integral over rectangles is the direct ancestor of every volume, mass, moment-of-inertia, and probability computation done with multiple integrals in applied mathematics, physics, and engineering — for instance computing the total mass of a rectangular plate with variable density $\rho(x,y)$ , or the flux of a field through a rectangular aperture. It is the base case from which Fubini's Theorem reduces double and triple integrals to iterated single-variable integrals, which is how such integrals are actually computed in practice, and it is the base case from which Integrals over General Regions extends integration to disks, balls, and arbitrary bounded domains by extending the integrand by zero outside the region and integrating over an enclosing rectangle. In probability and statistics, joint probability density functions over rectangular (or box-shaped, in higher dimensions) sample spaces are integrated exactly via this machinery to compute joint probabilities and expectations. In numerical analysis, the partition/Riemann-sum picture is the conceptual seed of numerical quadrature schemes in higher dimensions (e.g. tensor-product Simpson's rule, midpoint rule cubature), which approximate multiple integrals by refined finite sums exactly analogous to $U(f,P)$ and $L(f,P)$ . In computer graphics and finite-element methods, discretizing a domain into rectangular (or box) elements and summing local contributions is a direct computational descendant of upper/lower Riemann sums. The theoretical criterion also foreshadows Lebesgue's integrability criterion, which characterizes Riemann integrability precisely in terms of the set of discontinuities having Lebesgue measure zero — a refinement of the continuity-based sufficient condition proved here.

Ask the tutor: How does the proof that continuous functions are integrable change if the rectangle $R$ is replaced by a more general compact region?

R = [a_1,b_1] \times [a_2,b_2] \times \cdots \times [a_n,b_n] \subset \mathbb{R}^n

Intuition

Picture $n=2$ : a rectangle $R = [a_1,b_1]\times[a_2,b_2]$ in the plane, and a surface $z = f(x,y)$ sitting above it. A partition slices $R$ into a checkerboard grid of small rectangles $Q$ . Over each tiny rectangle, $f$ is approximately constant if $f$ is well-behaved, so the true volume under the surface over $Q$ is trapped between $m_Q(f)\,v(Q)$ (a flat box built to the lowest height reached on $Q$ ) and $M_Q(f)\,v(Q)$ (a flat box built to the highest height). Summing over all the tiles gives $L(f,P) \le (\text{true volume}) \le U(f,P)$ for every partition $P$ — this sandwiching is the entire content of the definition. Refining a partition (adding more grid lines in either direction) can only push $U(f,P)$ down and $L(f,P)$ up, because breaking a rectangle into smaller pieces can only shrink each local supremum and grow each local infimum (or leave them the same). The upper integral is the best (smallest) upper sum achievable; the lower integral is the best (largest) lower sum achievable. Integrability says these two "best bounds" meet exactly — the boxy over- and under-approximations of the volume converge to a single honest number as the grid becomes infinitely fine in every direction. The genuinely new phenomenon in $\mathbb{R}^n$ , compared to $\mathbb{R}^1$ , is that the grid must refine along all $n$ axes at once — one cannot simply refine in the $x$ -direction and declare victory while the $y$ -direction stays coarse. The criterion below packages exactly the condition needed to control this multi-directional refinement.

:::svg[A partition of $R=[0,2]\times[0,1]$ into 6 sub-rectangles, with one tile's sup/inf sandwich indicated]

R \subset \mathbb{R}^n

R \subset \mathbb{R}^n

L(f,P^*) \le U(f,P^*)

f: R \to \mathbb{R}

P^*

\Rightarrow

R \subset \mathbb{R}^n

R

Worked examples

Example 1: A genuinely two-dimensional continuous function, computed from the definition. Let $R = [0,1] \times [0,2] \subset \mathbb{R}^2$ and $f(x,y) = x + 2y$ . This is continuous, hence integrable by the theorem above; we verify the value using Riemann sums directly (without invoking Fubini, which is not yet available). Partition $[0,1]$ into $n$ equal sub-intervals of length $1/n$ and $[0,2]$ into $n$ equal sub-intervals of length $2/n$ , giving $n^2$ sub-rectangles

Q_{jk} = \left[\tfrac{j-1}{n}, \tfrac{j}{n}\right] \times \left[\tfrac{2(k-1)}{n}, \tfrac{2k}{n}\right], \qquad j,k = 1,\dots,n,

each of volume $v(Q_{jk}) = \dfrac{1}{n}\cdot\dfrac{2}{n} = \dfrac{2}{n^2}$ . Since $f(x,y) = x+2y$ is increasing in both $x$ and $y$ , its maximum on $Q_{jk}$ occurs at the corner with largest $x$ and largest $y$ , namely $(j/n,\, 2k/n)$ , and its minimum at $(\,(j-1)/n,\ 2(k-1)/n\,)$ . So

M_{Q_{jk}} = \frac{j}{n} + 2\cdot\frac{2k}{n} = \frac{j + 4k}{n}, \qquad m_{Q_{jk}} = \frac{j-1}{n} + 2\cdot\frac{2(k-1)}{n} = \frac{(j-1)+4(k-1)}{n}.

Upper sum:

U(f,P_n) = \sum_{j=1}^n \sum_{k=1}^n \frac{j+4k}{n}\cdot \frac{2}{n^2} = \frac{2}{n^3}\sum_{j=1}^n\sum_{k=1}^n (j + 4k).

Compute the double sum:

\sum_{j=1}^n\sum_{k=1}^n (j+4k) = \sum_{j=1}^n\left( nj + 4\cdot\frac{n(n+1)}{2}\right) = n\cdot\frac{n(n+1)}{2} + 2n^2(n+1) = \frac{n^2(n+1)}{2} + 2n^2(n+1).

= n^2(n+1)\left(\frac12 + 2\right) = \frac{5n^2(n+1)}{2}.

U(f,P_n) = \frac{2}{n^3}\cdot \frac{5n^2(n+1)}{2} = \frac{5(n+1)}{n} = 5 + \frac{5}{n}.

Lower sum: replace $j \to j-1$ , $k\to k-1$ in the roles above; by the same computation shifted,

\sum_{j=1}^n\sum_{k=1}^n \big((j-1)+4(k-1)\big) = \sum_{j=0}^{n-1}\sum_{k=0}^{n-1}(j+4k) = \frac{5(n-1)^2 n}{2}\cdot\frac{1}{n}\Big|_{\text{redo directly:}}

Let's redo cleanly: with $j' = j-1, k'=k-1$ ranging over $0,\dots,n-1$ ,

\sum_{j'=0}^{n-1}\sum_{k'=0}^{n-1}(j'+4k') = \sum_{j'=0}^{n-1}\left(n j' + 4\cdot\frac{(n-1)n}{2}\right) = n\cdot\frac{(n-1)n}{2} + 2n(n-1)n = \frac{n^2(n-1)}{2} + 2n^2(n-1)

= n^2(n-1)\left(\frac12+2\right) = \frac{5n^2(n-1)}{2}.

L(f,P_n) = \frac{2}{n^3}\cdot \frac{5n^2(n-1)}{2} = \frac{5(n-1)}{n} = 5 - \frac{5}{n}.

As $n\to\infty$ , $U(f,P_n) \to 5$ and $L(f,P_n) \to 5$ , and $U(f,P_n) - L(f,P_n) = 10/n \to 0$ , consistent with Riemann's Criterion. Both sequences squeeze to $5$ , so

\int_R (x+2y)\, dx\,dy = 5.

(Sanity check by elementary geometry: $\int_R f = \int_R x\,dA + 2\int_R y \, dA$ . The average value of $x$ over $[0,1]$ is $1/2$ and area of $R$ is $2$ , contributing $\tfrac12\cdot 2 = 1$ ; the average value of $y$ over $[0,2]$ is $1$ , contributing $2\cdot 1 \cdot 2 = 4$ . Total $1+4=5$ , matching.)

Example 2: Riemann's Criterion applied to a discontinuous function. Let $R = [0,1]\times[0,1]$ and define

f(x,y) = \begin{cases} 1 & \text{if } x \text{ is rational} \\ 0 & \text{if } x \text{ is irrational,} \end{cases}

independent of $y$ (a two-dimensional analogue of Dirichlet's function, discontinuous at every point). We show directly, via Riemann's Criterion, that $f$ is not integrable on $R$ — illustrating that the criterion is a genuine tool for disproving integrability, not just proving it. Let $P = (P_1,P_2)$ be any partition of $R$ , with $P_1$ dividing $[0,1]$ into sub-intervals $I_1,\dots,I_p$ and $P_2$ dividing $[0,1]$ into $J_1,\dots,J_q$ . Each sub-rectangle is $Q_{ab} = I_a \times J_b$ . Every non-degenerate interval $I_a$ (having positive length) contains both rational and irrational points (density of $\mathbb{Q}$ and $\mathbb{R}\setminus\mathbb{Q}$ in $\mathbb{R}$ ), so

M_{Q_{ab}}(f) = \sup_{Q_{ab}} f = 1, \qquad m_{Q_{ab}}(f) = \inf_{Q_{ab}} f = 0

for every $a,b$ . Hence

U(f,P) = \sum_{a,b} 1\cdot v(Q_{ab}) = v(R) = 1, \qquad L(f,P) = \sum_{a,b} 0 \cdot v(Q_{ab}) = 0.

This holds for every partition $P$ , so $\overline{\int_R} f = \inf_P U(f,P) = 1$ while $\underline{\int_R} f = \sup_P L(f,P) = 0$ . Since $1 \neq 0$ , Riemann's Criterion fails for every $\varepsilon < 1$ : no partition can make $U(f,P) - L(f,P) < \varepsilon$ once $\varepsilon \le 1$ is impossible to beat, and indeed $U(f,P)-L(f,P) = 1$ always. So $f$ is not Riemann integrable on $R$ .

f: R \to \mathbb{R}

Scope, edge cases & common traps

Situation	What holds	What can fail
$f$ continuous on closed rectangle $R$	Always integrable (theorem above)	—
$f$ bounded with finitely many discontinuities on $R$ (n=1 case, or measure-zero discontinuity sets in general)	Still integrable (a stronger theorem, via Lebesgue's criterion, not proved here)	Requires the discontinuity set to be "small" (measure zero)
$f$ bounded but discontinuous on a "thick" set (e.g. Example 2's Dirichlet-type function)	—	Integrability can fail entirely
$f$ unbounded on $R$	—	Riemann integral is not even defined; see counterexample
$R$ not closed or not bounded (e.g. open or infinite rectangle)	—	Compactness argument fails; extreme values may not be attained, uniform continuity may fail
Refining a partition	Upper sums decrease (weakly), lower sums increase (weakly)	Common refinement is essential to compare two different partitions meaningfully

Common trapThe classic error: confusing "some partition works" with "every fine-enough partition works"

Riemann's Criterion is often mis-stated with the wrong quantifier: students sometimes write " $f$ is integrable if $U(f,P) - L(f,P) < \varepsilon$ for the partition into $n$ equal pieces, as $n\to\infty$ ." This is not what the criterion says, and using equal-width grids is a special (though common) choice, not a logical necessity. The correct quantifier structure is: for every $\varepsilon>0$ there exists some partition $P$ (possibly wildly non-uniform, more refined in regions where $f$ oscillates) achieving $U(f,P)-L(f,P)<\varepsilon$ . Getting this existential/universal order backwards — e.g. claiming one must check all partitions shrink the gap, or that uniform partitions are the only ones that need testing — produces false proofs of both integrability and non-integrability. Example 2 above is precisely a case where every partition, uniform or not, gives the same gap of exactly $1$ ; recognizing that the argument must apply to an arbitrary partition (not just equal subdivisions) is what makes the non-integrability proof airtight.

The mathematician behind it

Origins · 1 minBernhard Riemann and how this was discovered

Bernhard Riemann

1826–1866 · 19th-century Göttingen, foundations of analysis and geometry

Georg Friedrich Bernhard Riemann was a German mathematician whose 1854 habilitation lecture and related work laid the rigorous groundwork for what is now called the Riemann integral, generalizing Cauchy's earlier integral for continuous functions to a broader class defined via oscillation of a function on partition elements. Riemann's approach — controlling the difference between upper-type and lower-type approximating sums — is exactly the mechanism formalized in the upper/lower sum definitions used here, later sharpened into the modern Darboux formulation by Jean-Gaston Darboux. Riemann's short life produced foundational results across analysis, differential geometry (Riemannian manifolds), and number theory (the Riemann zeta function and hypothesis), and his integral remains the standard first rigorous notion of integration taught before measure-theoretic integration.

History

- 1854 — Riemann's habilitation thesis at Göttingen introduces a rigorous integrability criterion for functions with possibly infinitely many discontinuities.

- 1875 — Jean-Gaston Darboux reformulates Riemann's integral using upper and lower sums (the presentation used in this article), clarifying the original oscillation-based definition.

- 1902 — Henri Lebesgue introduces measure-theoretic integration, later yielding the precise characterization (Lebesgue's criterion) of which bounded functions are Riemann integrable, in terms of the measure-zero size of the discontinuity set.

- Early 20th century — the Riemann integral over rectangles in

\mathbb{R}^n

, alongside Fubini's theorem (1907, Guido Fubini) for reducing multiple integrals to iterated one-variable integrals, becomes standard in multivariable calculus curricula.

- Mid-to-late 20th century — the rectangle-based Riemann/Darboux construction becomes the universal pedagogical entry point to multivariable integration, with extension to general (non-rectangular) regions handled by extending functions by zero.

What exactly is a "sub-rectangle" and why must the sides be partitioned independently?

A sub-rectangle is one cell of the grid formed by choosing one sub-interval from each side's partition and taking their Cartesian product. Partitioning each side independently (rather than, say, choosing a single mesh size for the whole rectangle) is what allows the grid to be refined unevenly — finer in directions or regions where $f$ oscillates more, coarser elsewhere. This flexibility is essential for the "for every $\varepsilon$ there exists a partition" structure of Riemann's Criterion: sometimes no uniform grid works efficiently, but a cleverly shaped one does. In $n$ dimensions the sub-rectangles tile $R$ exactly, with pairwise disjoint interiors, which is what makes volumes additive across the partition — a fact used repeatedly in the refinement lemma and the continuity theorem.

Is the upper integral always at least as large as the lower integral, even for non-integrable functions?

Yes. This is exactly the Corollary proved from the Refinement Lemma: for any bounded $f$ on a rectangle $R$ , one always has $\underline{\int_R} f \le \overline{\int_R} f$ , regardless of whether the two sides are equal. The proof only uses that any two partitions have a common refinement, and refining pushes lower sums up and upper sums down. Equality of the two sides is the extra condition that defines integrability; the inequality itself is unconditional and holds even for wild functions like the Dirichlet-type example, where it becomes the strict inequality $0 < 1$ .

Does continuity everywhere except at a few points still guarantee integrability?

Yes, but this is a stronger and separate theorem not proved in this article: a bounded function on a closed rectangle is Riemann integrable if its set of discontinuities has Lebesgue measure zero (Lebesgue's criterion for Riemann integrability). Continuity everywhere, proved here, is the special case where the discontinuity set is empty. Finitely many discontinuities, or discontinuities along finitely many line segments in $\mathbb{R}^2$ , still constitute a measure-zero set, so such functions remain integrable — but this requires measure-theoretic machinery beyond the compactness/uniform-continuity argument given here.

How does this definition connect to the later Fubini's Theorem?

This article defines $\int_R f$ as a single number obtained directly from $n$ -dimensional partitions and sums — it says nothing yet about computing that number via nested single-variable integrals. Fubini's Theorem is the separate (and computationally crucial) result that, under mild hypotheses, this $n$ -dimensional integral equals an iterated integral, e.g. $\int_R f = \int_{a_1}^{b_1}\!\left(\int_{a_2}^{b_2} f(x,y)\,dy\right) dx$ for $n=2$ . Riemann's Criterion proved here is typically a key ingredient in proving Fubini's Theorem, because one needs to know the multiple integral exists before showing it matches the iterated one.

Why can't we just define the integral using only one partition, say the "obvious" uniform grid?

Using only uniform grids and taking a limit as the mesh size shrinks to $0$ does, in fact, give the same integral when $f$ is integrable — this is a standard equivalent formulation. But the definition via upper/lower sums over all partitions is more fundamental because it directly produces a genuine number (the upper integral, an infimum, and lower integral, a supremum) without needing to first prove a limit exists. It also makes the non-integrability proofs (like Example 2) airtight: showing some clever partition fails is not enough to prove non-integrability, but showing every partition gives the same fixed gap is exactly what the all-partitions definition demands and what Example 2 verifies.

Does the theorem require

R

to be a "rectangle" in the strict sense, or does it work for other shapes?

The construction here is specific to rectangles (products of closed intervals) because partitioning each side independently is what produces the grid of sub-rectangles with cleanly additive volumes. For more general bounded regions $D \subset \mathbb{R}^n$ , one typically extends the integrand $f$ by zero to a rectangle $R \supseteq D$ and integrates the extended function over $R$ — this is exactly the strategy used in Integrals over General Regions, and it relies on the rectangle theory developed here as its foundation.

What role does compactness play beyond just "

R

is closed and bounded"?

Compactness (via Compactness in R^n) is used twice in the continuity theorem: first, to guarantee that continuous $f$ is uniformly continuous on $R$ (not merely continuous at each point), which is what lets a single $\delta$ work for the entire rectangle simultaneously; second, to guarantee that $f$ attains an actual maximum and minimum on each closed sub-rectangle $Q$ (extreme value theorem), so that $M_Q(f)$ and $m_Q(f)$ are honest function values whose difference can be bounded using uniform continuity. Without compactness, both of these could fail, as illustrated by unbounded or open domains where suprema/infima are not attained or are infinite.