Chapter 5: Integrals

Students explore the definite integral through the limit of Riemann sums, understanding how partitioning an interval and summing function values times subinterval widths approximates area. The Fundamental Theorem of Calculus is presented in two parts: connecting differentiation and integration, and showing how definite integrals can be evaluated using antiderivatives. Applications include calculating net change, determining displacement and distance traveled from velocity functions, and computing areas between curves. The chapter covers the interpretation of integrals in physical, biological, and economic contexts, such as total accumulated growth, work, and cost. Properties of definite integrals—such as additivity over intervals, reversal of limits, and the impact of constant factors—are emphasized for efficiency in problem solving. The concept of average value of a function is introduced, linking integration with mean quantities. Throughout, graphical intuition is combined with algebraic precision, preparing students for more advanced integration techniques in later chapters.