Fundamental Theorem of Calculus: Integrals and Primitives

This lecture covers the Fundamental Theorem of Calculus, focusing on the relationship between differentiation and integration. The instructor begins by discussing the concept of integrals and how they relate to the area under curves. The lecture emphasizes the importance of finding primitives, or antiderivatives, of functions, and how these can be used to compute definite integrals. The instructor illustrates this with examples, including constant functions and linear functions, demonstrating how to derive their primitives. The discussion progresses to more complex functions, such as polynomials and trigonometric functions, highlighting the techniques for finding their integrals. The lecture also addresses the significance of continuity in functions and the implications for their integrability. The instructor concludes by reinforcing the idea that every continuous function has a primitive, which is a key takeaway from the theorem. This foundational knowledge is essential for further studies in calculus and analysis, providing students with the tools to tackle more advanced topics in mathematics.