Definite Integral: Subdivisions and Finesse

This lecture focuses on the concept of definite integrals, approaching the area under a curve with positive and negative parts through partitions and subdivisions. The instructor explains the importance of regular partitions and the role of the step size in determining the finesse of the partition. By adjusting the step size, one can improve the accuracy of the definite integral approximation and understand the error involved. The lecture emphasizes the significance of the maximum distance between partition points in assessing the overall finesse. The instructor also introduces the concept of Darboux sums, highlighting their relevance in analysis and providing a theoretical foundation for numerical integration under a curve.