Lecture
This lecture focuses on Lagrange interpolation and its application in numerical integration. The instructor begins by discussing the importance of approximating definite integrals using numerical methods. They introduce the concept of partitioning intervals and using polynomials to approximate areas under curves. The lecture covers both non-composite and composite methods of quadrature, explaining how non-composite methods replace integrals with polynomial approximations over the entire interval, while composite methods involve partitioning the interval into subintervals for more accurate integration. The instructor emphasizes the significance of polynomial bases, particularly Lagrange polynomials, in constructing these approximations. The lecture also highlights the linear independence of these polynomials and their role in forming a basis for polynomial spaces. Throughout the session, practical examples and visualizations are provided to illustrate the concepts, ensuring that students grasp the underlying principles of numerical integration and interpolation techniques effectively.