Multiple Integrals

The one-variable Riemann integral partitions an interval $[a,b]$ and squeezes the area under a graph between lower and upper sums. Over a closed rectangle in ${\mathbb{R}}^n$ the same construction runs with the interval replaced by a box and its subintervals by a grid of subrectangles. A bounded $f:R\to \mathbb{R}$ is integrable when the two sums can be brought arbitrarily close, and the number they trap is ${\iint }_{R}f$.¹

Rectangles, partitions, and Darboux sums

A closed rectangle in ${\mathbb{R}}^n$ is a product of closed intervals $R=[a_1,b_1]\times \cdots \times [a_n,b_n]$. Partitioning each edge $[a_i,b_i]$ into finitely many subintervals cuts $R$ into a grid of subrectangles; the collection of these is a partition $P$ of $R$.

For the two-dimensional case $n=2$, a subrectangle is a cell $R_{\alpha }=[x_{i-1},x_i]\times [y_{j-1},y_j]$ of area $v(R_{\alpha })=\Delta x_i\Delta y_j$, and the upper sum stacks a box of height $M_{\alpha }$ on each cell, the lower sum a box of height $m_{\alpha }$. The graph of $f$ is caught between the two stacks.

Since $m_{\alpha }\le M_{\alpha }$ on every cell, $L(f,P)\le U(f,P)$ for a single partition. Refining $P$ — adding cut planes — can only raise lower sums and lower upper sums, exactly as in one variable, because splitting a cell replaces one $\sup$ (or $\inf$) by cell-wise suprema that fit more tightly.² Comparing any two partitions through their common refinement gives the fundamental inequality: every lower sum is at most every upper sum. The two families therefore have a gap that no partition crosses.

The integrability criterion

Testing every partition against the two integrals is unwieldy. The working test is the same squeeze that serves in one variable: integrability is the existence of a single partition making the sums close.

The criterion isolates the object to control: the gap ${\sum }_{\alpha }(M_{\alpha }-m_{\alpha })v(R_{\alpha })$, the total volume of the boxes trapped between the two stacks. Two ways of forcing this gap small are the source of every integrability theorem below — the oscillation $M_{\alpha }-m_{\alpha }$ is uniformly small where $f$ is continuous, and the cells where it is not can be given negligible total volume.

Content zero and continuous integrands

A set that a partition can bury under cells of arbitrarily small total volume contributes nothing to the gap. This is the higher-dimensional analogue of a finite set of jump points in one variable.

A single point has content zero, as does any finite union of content-zero sets. The graph of a continuous function on $[a,b]$ has content zero in ${\mathbb{R}}^2$: uniform continuity keeps the graph inside a chain of thin rectangles of total area $(b-a)$ times an oscillation that shrinks with the mesh. So the smooth boundary curves that bound the regions of the next section are negligible.

The continuous case follows from the Riemann criterion. A continuous function on the compact rectangle $R$ is uniformly continuous, so given $\epsilon >0$ there is a mesh fine enough that $M_{\alpha }-m_{\alpha }<\epsilon /v(R)$ on every cell. Summing, $U(f,P)-L(f,P)<\epsilon$. When the discontinuities have content zero, cover them by cells of small total volume — where $f$ oscillates by at most its global bound — and apply uniform continuity on the compact remainder; both contributions to the gap are small.

Fubini's theorem

Computing ${\iint }_{R}f$ from Darboux sums is impractical. Fubini's theorem reduces a multiple integral to nested one-variable integrals, evaluated in either order.

Each inner integral integrates $f$ along one direction, leaving a continuous function of the free variable that the outer integral then integrates. Geometrically, ${\int }_c^{d}f(x,y)\mathbf{d}y$ is the area of the slice of the solid under the graph cut by the plane at a fixed $x$, and the outer integral sweeps those slice-areas across $[a,b]$ to accumulate the volume.

Continuity is what makes the reduction clean: for continuous $f$ the inner integral $x\mapsto {\int }_c^{d}f(x,y)\mathbf{d}y$ is itself a continuous function of $x$, so the outer integral is an ordinary one-variable Riemann integral. For merely integrable $f$ the iterated integrals still compute ${\iint }_{R}f$ when they exist, but the inner integral can fail to exist at isolated values of $x$, and the two orders need separate justification.

Properties of the integral

The integral over a rectangle inherits, cell by cell from the Darboux sums, the structural properties of the one-variable integral:

• Linearity. ${\iint }_R(\alpha f+\beta g)=\alpha {\iint }_{R}f+\beta {\iint }_{R}g$ for scalars $\alpha ,\beta$.
• Monotonicity. $f\le g$ on $R$ forces ${\iint }_{R}f\le {\iint }_{R}g$; taking $g=|f|$ gives $\left|{\iint }_{R}f\right|\le {\iint }_R|f|$.
• Additivity. A cut plane splitting $R$ into $R^{\prime }$ and $R^{\prime \prime }$ splits the integral, ${\iint }_{R}f={\iint }_{R^{\prime }}f+{\iint }_{R^{\prime \prime }}f$, and the same holds for two regions meeting only along a set of content zero.

Each relation holds on every subrectangle and is preserved on passing to the infimum and supremum over partitions. Additivity is what lets a complicated region be integrated by cutting it into pieces on which one order of iteration is simple.

Regions bounded by curves

Rectangles are rarely the domain of interest. The integral over a general bounded region $D$ is defined by extending $f$ to be zero outside $D$ and integrating the extension over any rectangle $R\supseteq D$.

The extension $\tilde{f}$ can be discontinuous only where $f$ is discontinuous or on the boundary $\partial D$, where the indicator jumps. When $\partial D$ has content zero — true for a region bounded by finitely many graphs of continuous functions — the discontinuity set of $\tilde{f}$ has content zero, so a continuous $f$ on such a $D$ is integrable.

For a region between two curves, Fubini reduces the double integral to an iterated one whose inner limits are the bounding curves. If $D=\{(x,y):a\le x\le b,{\phi }_1(x)\le y\le {\phi }_2(x)\}$ with ${\phi }_1,{\phi }_2$ continuous, then

and symmetrically for a region described by ${\psi }_1(y)\le x\le {\psi }_2(y)$. When a region admits both descriptions, either order computes the same integral, and one order is often far easier than the other.

Summary

The multiple integral over a rectangle is the one-variable Darboux construction with subrectangles in place of subintervals; the squeeze criterion, content-zero negligible sets, and integrability of continuous functions all carry over. Fubini's theorem turns the integral into iterated one-variable integrals in either order, and the zero-extension carries the theory to regions bounded by continuous curves.