Numerical Integration: Lagrange Interpolation Methods

This lecture discusses numerical integration techniques, focusing on Lagrange interpolation. The instructor introduces the concept of approximating integrals using quadrature methods, specifically the midpoint rule and Riemann sums. The lecture explains the mean value theorem for integrals, illustrating how to express definite integrals in terms of function values at specific points. The instructor emphasizes the importance of selecting appropriate points for approximation, detailing methods such as the left point, right point, and trapezoidal rule. The discussion progresses to Lagrange interpolation, where the goal is to find a polynomial that passes through given data points. The instructor outlines the uniqueness of the polynomial and its construction through linear combinations of basis polynomials. The lecture concludes with practical examples and visualizations to demonstrate how these methods can be applied to approximate integrals and construct interpolating polynomials effectively.