A tool that numerically estimates the definite integral of a function by partitioning the interval of integration into subintervals and evaluating the function at the right endpoint of each subinterval. The area of each rectangle formed by this height and the subinterval width is then calculated, and the sum of these areas provides an approximation of the integral’s value. For example, to approximate the integral of f(x) = x2 from 0 to 2 using 4 subintervals, the function would be evaluated at x = 0.5, 1, 1.5, and 2. The approximation is then (0.52 0.5) + (12 0.5) + (1.52 0.5) + (22 0.5) = 3.75.
The utility of such a calculation lies in its ability to approximate definite integrals of functions that lack elementary antiderivatives or when only discrete data points are available. Its historical context stems from the fundamental development of integral calculus, where methods for approximating areas under curves were crucial before the establishment of analytical integration techniques. The benefits of using such a method include its simplicity and applicability to a wide range of functions, providing a reasonable estimate of the definite integral, especially when the number of subintervals is sufficiently large.
