What does the value of an integral represent?

The value of an integral represents the area under a curve on a given interval. When you compute the definite integral of a function from one point to another, you are essentially calculating the total accumulated area that lies between the curve of the function and the x-axis within that interval.

This concept is a fundamental principle in calculus, as it allows us to quantify how much "space" is enclosed by the function and the x-axis, which is particularly useful in diverse applications, such as physics for calculating distances and in economics for finding total revenue or cost over a period.

While the other options point to different aspects related to functions, they do not capture the primary interpretation of an integral. The slope of a function is represented by its derivative, the maximum value would pertain to the evaluation of a function at its critical points, and the average value involves different calculations entirely, usually framed in terms of integrals but focused on a mean concept rather than the area itself. Thus, the definition of the integral closely aligns with the area interpretation, making it the correct choice.