Multiple integral on a rectangular region

We extend the notion of integral to multivariate functions. For a bivariate function $f(x,y)$ on a closed region $D$, its double integral

$${\iint }_{D}f(x,y)dxdy$$

is defined similarly to what we did for univariate functions, using the Riemann sums.

In this and the following posts, we mainly deal with integrating bivariate (two-variable) functions (i.e., double integral). Roughly speaking, a double integral corresponds to the volume of a solid between a region on the $x$-$y$ plane and a two-variable function $z=f(x,y)$ in ${\mathbb{R}}^3$.

Riemann sums

The integral of multivariate functions can be defined in much the same manner as univariate functions. In the case of univariate functions, we first considered integrals on a closed interval. In the case of bivariate functions, we first consider integrals on a bounded closed region, but the region can be of any shape. We start with a rectangular region.

Let $f(x,y)$ be a bounded function on the rectangular region $D=[a,b]\times [c,d]$. We define a partition of $D$ to be the pair of the partition of the closed interval $[a,b]$ and the partition of the closed interval $[c,d]$:

$$\mathrm{\Delta }:\{\begin{array}{c}a=a_0<a_1<a_2<\cdots <a_{n-1}<a_n=b,\\ c=c_0<c_1<c_2<\cdots <c_{m-1}<c_m=d.\end{array}$$

By this partition $\mathrm{\Delta }$, the rectangular region $D$ is partitioned into $nm$ small rectangular regions:

$$D_{ij}=[a_i,a_{i+1}]\times [c_j,c_{j+1}]$$

for each $i=0,1,\cdots ,n-1$ and each $j=0,1,\cdots ,m-1$. For each of these small regions, let us define the following:

$$\begin{array}{rcl}{M}_{ij}& =& sup\{f(x,y)\mid (x,y)\in D_{ij}\},\\ m_{ij}& =& inf\{f(x,y)\mid (x,y)\in D_{ij}\}.\end{array}$$

That is, $M_{ij}$ and $m_{ij}$ are the supremum and infimum, respectively, of $f(x,y)$ in each rectangular region $D_{ij}$. The sum of the volumes of the quadrangular prisms with base $D_{ij}$ and height $M_{ij}$, given by

$$S_{\mathrm{\Delta }}=\sum _{i=0}^{n-1}\sum _{j=0}^{m-1}{M}_{ij}(a_{i+1}-a_i)(c_{j+1}-c_j),$$

is called the upper Riemann sum of $f(x,y)$ with respect to the partition $\mathrm{\Delta }$. Similarly,

$$s_{\mathrm{\Delta }}=\sum _{i=0}^{n-1}\sum _{j=0}^{m-1}{m}_{ij}(a_{i+1}-a_i)(c_{j+1}-c_j)$$

is called the lower Riemann sum of $f(x,y)$ with respect to the partition $\mathrm{\Delta }$.

Just as in the case of univariate functions, $S_{\mathrm{\Delta }}$ and $s_{\mathrm{\Delta }}$ are monotone decreasing, and increasing, respectively, with refinements of $\mathrm{\Delta }$.

We say the partition ${\mathrm{\Delta }}^{\prime }$ is a refinement of the partition $\mathrm{\Delta }$ if the partitions of $[a,b]$ and $[c,d]$ in ${\mathrm{\Delta }}^{\prime }$ are refinements of the partitions of $[a,b]$ and $[c,d]$ in $\mathrm{\Delta }$. In other words, each small rectangle in $\mathrm{\Delta }$ is partitioned into smaller rectangles in ${\mathrm{\Delta }}^{\prime }$. We have

$$S_{\mathrm{\Delta }}\ge S_{{\mathrm{\Delta }}^{\prime }},s_{\mathrm{\Delta }}\le s_{{\mathrm{\Delta }}^{\prime }}.$$

For a sequence of refinements of partitions $\mathrm{\Delta },{\mathrm{\Delta }}^{\prime },{\mathrm{\Delta }}^{″},\cdots$, we have (as in the case of univariate functions),

$$s_{\mathrm{\Delta }}\le s_{{\mathrm{\Delta }}^{\prime }}\le s_{{\mathrm{\Delta }}^{″}}\le \cdots S_{{\mathrm{\Delta }}^{″}}\le S_{{\mathrm{\Delta }}^{\prime }}\le S_{\mathrm{\Delta }}.$$

Thus, $\underset{\mathrm{\Delta }}{sup}{s}_{\mathrm{\Delta }}$ (the supremum of lower Riemann sums over all possible partitions) and $\underset{\mathrm{\Delta }}{inf}{S}_{\mathrm{\Delta }}$ (the infimum of upper Riemann sums over all possible partitions) exist. Clearly, the following holds:

$$\begin{array}{}\text{(Eq:supinfS)}& \underset{\mathrm{\Delta }}{sup}{s}_{\mathrm{\Delta }}\le \underset{\mathrm{\Delta }}{inf}{S}_{\mathrm{\Delta }}\end{array}$$

Definition (Definite (double) integral)

The function $f(x,y)$ that is bounded on the rectangular region $D=[a,b]\times [c,d]$ is said to be Riemann-integrable if

$$\underset{\mathrm{\Delta }}{sup}{s}_{\mathrm{\Delta }}=\underset{\mathrm{\Delta }}{inf}{S}_{\mathrm{\Delta }}.$$

Then, we write

$${\iint }_{D}f(x,y)dxdy=\underset{\mathrm{\Delta }}{sup}{s}_{\mathrm{\Delta }}(=\underset{\mathrm{\Delta }}{inf}{S}_{\mathrm{\Delta }})$$

and call it the definite (double) integral of $f(x,y)$ on $D$.

In the following, we simply say "integrable" to mean Riemann-integrable.

We can also define the triple integral of the form

$${\iiint }_{D}f(x,y,z)dxdydz$$

where $D=[a,b]\times [c,d]\times [e,f]$ is a cuboid region (a "box") in ${\mathbb{R}}^3$ (analog of a rectangular region in ${\mathbb{R}}^2$), and $f(x,y,z)$ is a 3-variable function. 

More generally, we can define the multiple integrals of the form

$$\int \cdots {\int }_{D}f(x_1,\cdots ,x_n)d{x}_1\cdots d{x}_n$$

where $D=[a_1,b_1]\times [a_2,b_2]\times \cdots \times [a_n,b_n]$ is a "box" in ${\mathbb{R}}^n$ and $f(x_1,x_2,\cdots ,x_n)$ is an $n$-variable function.

Properties of multiple integrals

From the above definition, the following theorems should be trivial.

Theorem (Integration is a linear operation)

Let $f(x,y)$ and $g(x,y)$ be integrable functions on a rectangular region $D$. Let $k,l\in \mathbb{R}$. Then, $kf(x,y)+lg(x,y)$ is also integrable on $D$, and we have

$${\iint }_D(kf(x,y)+lg(x,y))dxdy=k{\iint }_{D}f(x,y)dxdy+l{\iint }_{D}g(x,y)dxdy.$$

Theorem (Integration is additive)

Let $f(x,y)$ be an integrable function on a rectangular region $D$. Suppose $D$ is partitioned into a finite number of rectangular regions $D_1,D_2,\cdots ,D_r$. Then, $f(x,y)$ is integrable on each $D_i$, and we have

$${\iint }_{D}f(x,y)dxdy=\sum _{i=1}^r{\iint }_{D_i}f(x,y)dxdy.$$

Just as for univariate functions, we have the following theorem.

Theorem (Continuous functions are integrable)

Let $f(x,y)$ be a continuous function on a rectangular region $D=[a,b]\times [c,d]$. Then $f(x,y)$ is integrable on $D$.

Proof. Omitted. ■