A function defined by multiple sub-functions, each applying to a specific interval of the domain, requires a specialized approach when calculating the area under its curve within defined limits. The overall area is determined by summing the areas computed for each sub-function over its respective interval. For example, consider a function that equals x for x < 0 and x for x 0. To find the area under this curve from -1 to 1, the integral of x from -1 to 0 is added to the integral of x from 0 to 1.
This methodology provides a precise way to analyze systems modeled by distinct behaviors across different operating ranges. It allows for the exact determination of accumulated quantities where the rate of change is not uniform, facilitating accurate predictions and assessments in diverse fields such as physics, engineering, and economics. Its use dates back to the development of calculus, growing in application alongside the need for refined modeling techniques.
