Quadrature Formulas: Composite and Non-Composite Methods

This lecture discusses quadrature methods, focusing on the distinction between composite and non-composite techniques. The instructor explains how non-composite quadrature methods replace the integral of a function with an interpolating polynomial, while composite methods involve partitioning the interval and applying interpolation within subintervals. The lecture covers various quadrature formulas, including the midpoint and trapezoidal rules, and provides examples to illustrate their application. The instructor emphasizes the importance of understanding the exactness of these formulas for polynomials of different degrees. Additionally, the lecture introduces the concept of changing variables to simplify the integration process, leading to a standard interval for analysis. The instructor also defines quadrature formulas and discusses their approximation properties, highlighting the significance of weights and nodes in these methods. The lecture concludes with practical examples, demonstrating how to apply these quadrature methods effectively in various scenarios, including the Simpson's rule and its advantages for linear functions.