Which of the following describes an unbounded area?

An unbounded area is defined as an area under a curve that extends infinitely, meaning it does not have specified limits or boundaries. In the context of integration, when we consider the area under a curve, if the curve itself extends indefinitely in one or both directions, it indicates that the area is not confined to a finite region.

For instance, curves like ( y = \frac{1}{x} ) as ( x ) approaches 0 or as ( x ) approaches infinity illustrate situations where the area can keep growing without reaching a defined limit. Therefore, this characteristic of extending infinitely aligns with the definition of an unbounded area.

In contrast, the other options describe either finite areas or constrained situations. The area under a curve that approaches a finite limit refers to bounded areas, while areas contained within specified boundaries clearly denote regions that have both upper and lower limits. Additionally, the concept of approximation does not inherently define whether an area is bounded or unbounded but rather concerns the means by which we might estimate areas. Thus, the defining quality of extending infinitely firmly categorizes option B as the correct description of an unbounded area.