A numerical method for approximating the definite integral of a function is achieved by utilizing rectangles. This particular method, employing right endpoints, calculates the area of several rectangles constructed along the x-axis. The height of each rectangle is determined by the function’s value at the rightmost point within each subinterval. The sum of these rectangular areas then serves as an estimate of the total area under the curve of the function within the defined interval. For instance, to estimate the definite integral of f(x) = x from 0 to 2 using 4 subintervals, the height of each rectangle is f(0.5), f(1), f(1.5), and f(2) respectively. The width of each rectangle is 0.5, and summing the areas of these rectangles yields an approximation of the integral.
This approximation technique provides a computationally efficient way to estimate definite integrals, particularly when analytical solutions are difficult or impossible to obtain. It is fundamental in numerical analysis and finds application in various fields, including physics, engineering, and economics, where calculating the area under a curve representing a physical or economic quantity is essential. Historically, these summation methods predate modern calculus and represent early approaches to area and integration problems. Their utility extends to scenarios involving discrete data points where a continuous function is not explicitly defined.
