Ohio Assessments for Educators (OAE) Mathematics Practice Exam

Which of the following represents the area under a curve?

1. Derivative

2. Definite Integral

3. Limit

4. Continuous Function

The correct answer is: Definite Integral

The area under a curve is represented by the definite integral. This is a fundamental concept in calculus where the definite integral of a function over a specified interval gives the total area between the curve of the function and the x-axis within that interval. In mathematical terms, if you have a function f(x) defined on the interval [a, b], the definite integral of f from a to b is written as ∫[a to b] f(x) dx. This integral calculates the accumulation of values represented by the function, which geometrically corresponds to the area enclosed by the curve, the x-axis, and the vertical lines x = a and x = b. Other concepts mentioned, such as the derivative, limit, and continuous function, serve different purposes in calculus. While the derivative gives the slope or rate of change of a function, and limits are used to define derivatives and integrals, they do not directly represent the area under a curve. A continuous function, although it can have an area under its curve, is simply a property of functions that indicates they do not have breaks or jumps - it does not by itself measure area. Thus, the definite integral distinctly represents the area under a curve in the context of calculus.