Midpoint Rule
The Midpoint Rule is a numerical integration technique used to approximate the definite integral of a function by dividing the integration interval into subintervals and using the midpoint of each subinterval to estimate the area under the curve. It is a specific case of Riemann sums and is part of the broader family of Newton-Cotes formulas for numerical quadrature. This method is particularly useful when an analytical solution to an integral is difficult or impossible to obtain.
Developers should learn the Midpoint Rule when working on applications involving numerical analysis, scientific computing, or simulations that require integration, such as in physics engines, financial modeling, or data science for approximating probabilities. It is often preferred over simpler methods like the left or right Riemann sums because it typically provides better accuracy for smooth functions, making it a foundational skill for implementing efficient numerical algorithms in programming languages like Python, MATLAB, or C++.