{"id":"integral","domain":"analysis","type":"Concept","title":"Integral","slug":"integral","url":"/mathematics/analysis/integral/","summary":"Computes total accumulation, most famously area under a curve. Defined as a limit of Riemann sums; connected to the derivative via the Fundamental Theorem of Calculus.","formal_definition":"∫ₐᵇf(x)dx = lim(n→∞) Σf(xᵢ)Δx (Riemann sum limit)","key_results":[{"result":"Fundamental Theorem of Calculus","statement":"∫ₐᵇf(x)dx = F(b)-F(a) where F'=f"}],"added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Rudin, W. (1976). Principles of Mathematical Analysis. 3rd ed. McGraw-Hill."},{"tier":1,"citation":"Royden, H.L. and Fitzpatrick, P. (2010). Real Analysis. 4th ed. Pearson."},{"tier":2,"citation":"Spivak, M. (2008). Calculus. 4th ed. Publish or Perish."},{"tier":3,"citation":"Boyer, C.B. (1949). The History of the Calculus and its Conceptual Development. Dover."}],"relationships":[{"type":"built_from","target":"limit","label":"Limit"},{"type":"inverse_of","target":"derivative","label":"Derivative"},{"type":"anticipated_by","target":"archimedes","label":"Archimedes — method of exhaustion"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.4"}