Multiple integral on a rectangular region

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

\[\iint_Df(x,y)dxdy\]

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 \([c,d]\):

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

By this partition \(\Delta\), 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:

\[\begin{eqnarray} M_{ij} &=& \sup\{f(x,y)\mid (x,y) \in D_{ij}\},\\ m_{ij} &=& \inf\{f(x,y)\mid (x,y) \in D_{ij}\}. \end{eqnarray}\]

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

\[S_{\Delta} = \sum_{i=0}^{n-1}\sum_{j=0}^{m-1}M_{ij}(a_{i+1} - a_{i})(c_{j+1} - c_{j}),\]

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

\[s_{\Delta} = \sum_{i=0}^{n-1}\sum_{j=0}^{m-1}m_{ij}(a_{i+1} - a_{i})(c_{j+1} - c_{j})\]

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

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

We say the partition \(\Delta'\) is a refinement of the partition \(\Delta\) if the partitions of \([a,b]\) and \([c,d]\) in \(\Delta'\) are refinements of the partitions of \([a,b]\) and \([c,d]\) in \(\Delta\). In other words, each small rectangle in \(\Delta\) is partitioned into smaller rectangles in \(\Delta'\). We have

\[S_{\Delta} \geq S_{\Delta'}, ~ s_{\Delta} \leq s_{\Delta'}.\]

For a sequence of refinements of partitions \(\Delta, \Delta', \Delta'', \cdots\), we have (as in the case of univariate functions),

\[s_{\Delta} \leq s_{\Delta'} \leq s_{\Delta''} \leq \cdots S_{\Delta''} \leq S_{\Delta'} \leq S_{\Delta}.\]

Thus, \(\sup_{\Delta}s_{\Delta}\) (the supremum of lower Riemann sums over all possible partitions) and \(\inf_{\Delta}S_{\Delta}\) (the infimum of upper Riemann sums over all possible partitions) exist. Clearly, the following holds:

\[\sup_{\Delta}s_{\Delta} \leq \inf_{\Delta}S_{\Delta} \tag{Eq:supinfS}\]

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

\[\sup_{\Delta}s_{\Delta} = \inf_{\Delta}S_{\Delta}.\]

Then, we write

\[\iint_{D}f(x,y) dxdy = \sup_{\Delta}s_{\Delta} (= \inf_{\Delta}S_{\Delta})\]

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

\[\iiint_Df(x,y,z)dxdydz\]

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

\[\idotsint_Df(x_1, \cdots, x_n)dx_1\cdots dx_n\]

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

\[\iint_D(kf(x,y) + lg(x,y))dxdy = k\iint_Df(x,y)dxdy + l\iint_Dg(x,y)dxdy.\]

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

\[\iint_Df(x,y)dxdy = \sum_{i=1}^{r}\iint_{D_i}f(x,y)dxdy.\]

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. ■


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