Newton-Cotes Formulas
Newton-Cotes formulas are a group of numerical integration techniques used to approximate definite integrals by replacing the integrand with an interpolating polynomial and integrating that polynomial exactly. They are based on equally spaced points within the integration interval and include well-known methods like the trapezoidal rule and Simpson's rule. These formulas are fundamental in numerical analysis for solving integrals that are difficult or impossible to evaluate analytically.
Developers should learn Newton-Cotes formulas when working on scientific computing, engineering simulations, or data analysis tasks that require numerical integration, such as calculating areas under curves, solving differential equations, or processing signal data. They are particularly useful in fields like physics, finance, and machine learning where integrals arise frequently, and provide a straightforward approach with varying accuracy levels depending on the chosen rule (e.g., use Simpson's rule for higher precision over the trapezoidal rule).