Integration Summary

Real Analysis · Axiom Academy

How the Riemann integral formalizes area, connects to differentiation through the Fundamental Theorem of Calculus, and extends to improper integrals and the Gamma function. The integral is a limit of sums. The Riemann/Darboux integral pins down "area under a curve" rigorously: a bounded f is integrable on [a,b] exactly when its upper and lower Darboux sums can be squeezed together — . The integral behaves linearly and additively. Linearity, additivity over adjacent intervals, and monotonicity make the integral a well-behaved operator you can compute with. The FTC bridges the two halves of calculus. Part 1 says differentiating an accumulation function recovers the integrand; Part 2 turns integration into "evaluate an antiderivative at the endpoints." Techniques reduce hard integrals to easy ones. Substitution reverses the chain rule and integration by parts reverses the product rule. Improper integrals + comparison reach infinity. Limits extend integration to unbounded intervals and integrands; the comparison test decides convergence, and the Gamma and Beta functions are the headline payoff. Core Concept The Riemann / Darboux Integral Partition [a,b] , approximate the area by rectangles, and refine. The Darboux view brackets this with the lower sum L(f,P) (infimum heights) and upper sum U(f,P) (supremum heights). Integrability criterion: f is integrable for every there is a partition with .

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