Rectangular Regions and Their Side Lengths

Why Rectangular Regions Matter

A double integral is taken over a region in the plane. Before describing what the integral measures or how to evaluate one, it is important to specify the region precisely. For a rectangle whose sides are parallel to the coordinate axes, two intervals give a concise description: one interval specifies the allowed \(x\)-coordinates, and the other specifies the allowed \(y\)-coordinates.

Those bounds also determine the rectangle’s dimensions. The horizontal side length is the difference between the largest and smallest allowed \(x\)-coordinates; the vertical side length is the corresponding difference for \(y\). Their product gives the area. Keeping these ideas connected makes it easier to read and write the domain of a double integral.

Definition of a Coordinate Rectangle

The notation \([a,b]\) means all real numbers from \(a\) through \(b\), including both endpoints. Likewise, \([c,d]\) includes \(c\) and \(d\). The Cartesian product combines the intervals: a point \((x,y)\) belongs to \(D\) exactly when its first coordinate is in \([a,b]\) and its second coordinate is in \([c,d]\).

The inequalities \(a\le b\) and \(c\le d\) ensure that each interval is written from its smaller endpoint to its larger endpoint. When both inequalities are strict, the rectangle has positive width and height. If \(a=b\) or \(c=d\), one side length is zero, so the set is a line segment or a point rather than a rectangle with positive area.

Side Lengths and Area

The interval \([a,b]\) controls horizontal position. Its endpoints are \(a\) and \(b\), so the horizontal distance between them is \(b-a\). Similarly, the vertical distance between \(c\) and \(d\) is \(d-c\). The side lengths are distances, and are therefore nonnegative under the assumed ordering of the bounds.

This formula uses differences, not the coordinate values themselves. For example, the interval \([-2,4]\) has length \(4-(-2)=6\), not \(4\). A rectangle’s position in the plane does not determine its area; its side lengths do.

Worked Examples

Why the Area Formula Works

The coordinate description and the geometric dimensions come from the same bounds. The smallest and largest \(x\)-coordinates are \(a\) and \(b\). A horizontal segment joining the two vertical sides therefore spans a coordinate change of \(b-a\). In the same way, a vertical segment joining the lower and upper sides spans a coordinate change of \(d-c\).

These lengths are the rectangle’s width and height. The elementary area formula for a rectangle is width times height. Substituting the lengths determined by the coordinate intervals gives $$ \operatorname{Area}(D) =\text{width}\cdot\text{height} =(b-a)(d-c). $$ This also explains why shifting a rectangle without changing either side length leaves its area unchanged: the formula depends on the differences between endpoints, not on their absolute positions.

The set description follows directly from the meaning of a Cartesian product. A point \((x,y)\) is in \([a,b]\times[c,d]\) if and only if \(x\in[a,b]\) and \(y\in[c,d]\). Writing out those interval memberships gives \(a\le x\le b\) and \(c\le y\le d\). Hence the product notation and the inequality description specify exactly the same points.

Translating Bounds into an Integral Domain

When coordinate bounds are given, form the domain by pairing the \(x\)-interval with the \(y\)-interval. The conditions must hold simultaneously: a point is included only if it satisfies both coordinate restrictions. In the notation \([a,b]\times[c,d]\), the first interval always refers to \(x\), and the second refers to \(y\).

For example, the bounds \(-1\le x\le2\) and \(3\le y\le8\) define $$ D=[-1,2]\times[3,8] =\{(x,y)\in\mathbb{R}^2:-1\le x\le2,\ 3\le y\le8\}. $$ Its horizontal side length is \(2-(-1)=3\), its vertical side length is \(8-3=5\), and its area is \(3\cdot5=15\). Thus an expression such as \(\iint_D f(x,y)\,dA\) specifies that the function is integrated over precisely this rectangle. The domain notation describes where the points lie; it does not itself evaluate the integral.

A Common Mistake: Using an Endpoint as a Length

A frequent error is to treat a coordinate endpoint as if it were a side length. For the interval \([-2,4]\), one might incorrectly write \(4\) as the horizontal length and then calculate the area of \([-2,4]\times[1,5]\) as \(4\cdot4=16\). This is wrong because the horizontal interval begins at \(-2\), not at \(0\).

The correction is to subtract the lower endpoint from the upper endpoint: $$ 4-(-2)=6,\qquad 5-1=4,\qquad \operatorname{Area}(D)=6\cdot4=24. $$ A reliable check is to ask whether the proposed side length equals the distance between its two endpoints. The coordinate difference must be used even when the interval crosses zero or lies entirely below zero.

Why These Descriptions Matter

Writing a region accurately prevents ambiguity about which points belong to an integral’s domain. The product notation is compact, while inequalities make the inclusion conditions explicit. Both are useful, and translating between them is straightforward once the coordinate roles are clear.

Side lengths and area provide additional checks. If the bounds are ordered correctly, each side length is nonnegative. If lengths carry units, their product has square units. If the computed area is negative or the units remain linear, an arithmetic or interpretation error has occurred. These checks are especially useful before working with more elaborate descriptions of planar regions.