The process of calculating the area under a curve is a fundamental concept in calculus. This process extends to scenarios where the function defining the curve is not a single, continuous expression, but rather a collection of different expressions defined over specific intervals. For instance, a function might be defined as x2 for values of x less than 0, and as x for values of x greater than or equal to 0. Evaluating the accumulated area under such a function across a given interval requires dividing the integral into sub-integrals, one for each piece of the function within that interval. The final result is the sum of these individual integral values.
This approach is essential in numerous fields, including physics, engineering, and economics. In physics, it may be used to determine the work done by a force that varies in a piecewise manner. In engineering, it can assist in modeling systems with varying parameters. In economics, it may be applied to calculate total costs or revenues when different pricing strategies are in effect at different production levels. Historically, the need to analyze such scenarios motivated the development of techniques for handling such functions, allowing for more realistic and accurate modeling of real-world phenomena. This expands the applicability of integral calculus beyond purely continuous functions.
