A tool designed for numerical integration utilizes rectangles to approximate the area under a curve. The height of each rectangle corresponds to the function’s value at the left endpoint of the subinterval. For example, to approximate the area under f(x) from a to b using four rectangles, the area is calculated as h * [f(a) + f(a + h) + f(a + 2h) + f(a + 3h)], where h = (b – a) / 4 represents the width of each rectangle.
This method provides a relatively simple way to estimate definite integrals, particularly when an analytical solution is difficult or impossible to obtain. It plays a vital role in various fields, from computer graphics and physics simulations to financial modeling and data analysis, offering a practical approach to problems involving continuous change. Its historical context lies in the foundational work of Bernhard Riemann, whose contributions to calculus significantly advanced our understanding of integration.
