Closed Region on a Plane

We now begin the construction of the double integral, starting with a careful discussion of closed regions in the plane.

A closed region in the (x, y) plane is a set D of real points (x, y) given by inequalities

a₁ ≤ x ≤ a₂, b₁(x) ≤ y ≤ b₂(x),

where b₁(x) and b₂(x) are continuous and b₁(x) ≤ b₂(x) for x in [a₁, a₂] (Figure 12.1.3(a)). The boundary of D is the set of points in D which are on the curves

x = a₁, x = a₂, y = b₁(x), y = b₂(x).

Figure 12.1.3 (a) Closed region

The simplest type of closed region is a closed rectangle

a₁ ≤ x ≤ a₂, b₁ ≤ y ≤ b₂,

shown in Figure 12.1.3(b).

Figure 12.1.3 (b) Closed rectangle

Remark In this course we are restricting our attention to a very simple type of closed region, sometimes called a basic closed region. In advanced calculus and beyond, a much wider class of closed regions is studied.

An open region is a set of real points defined by strict inequalities of the form

c₁ < x < c₂, d₁(x) < y < d₂(x).

We shall usually be working with closed regions. So from now on when we use the word region alone we mean closed region.

To simplify our treatment we shall consider only continuous functions.

If f(x, y) ≥ 0 on D, the double integral is intuitively the volume of the solid over D between the surfaces z = 0 and z = f(x, y); i.e., the solid consisting of all points (x,y, z) where (x, y) is in D and 0 ≤ z ≤ f(x, y).

If f(x, y) ≤ 0 on D the double integral is intuitively the negative of the volume of the solid under D between the surfaces z = f(x, y) and z = 0. Thus volumes above the plane z = 0 are counted positively and volumes below z = 0 are counted negatively (Figure 12.1.4).

Figure 12.1.4