{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.integral","code":"thing-q80091","url":"https://ver.cy/models/thing/q80091/","name":"integral","alternateNames":["optical path length","absement","air mass","tabular integration","Harish-Chandra integral","Cauchy principal value","Birkhoff integral","Complete Fermi–Dirac integral","definite integral","Pettis integral","complex integral","partial fractions in integration"],"kind":"thing","status":"published","version":"0.2.0-wave.2","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QLT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"A fundamental concept of calculus with definite and indefinite forms: a definite integral assigns a number to a function over an interval or region when it exists, interpreted as area, accumulated change or total quantity, while an indefinite integral denotes a family of antiderivatives; generalisations include line, surface, multiple and Lebesgue integrals and principal values, and integrals appear across science as accumulated quantities like optical path length and absement.","purpose":"Let an agent explain integrals and integration techniques, compute and check integrals, interpret them in applications, and explain generalisations.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Accumulation operation","Inverse of differentiation","Definite and indefinite","Many generalisations"],"structure":{"bundles":[{"id":"compute","name":"Compute","description":"Evaluating integrals.","layers":[{"id":"techniques","name":"Techniques","description":"Methods.","findings":[{"id":"techniques-finding","name":"Techniques","description":"Methods.","questions":[{"text":"Which technique, such as substitution, parts or partial fractions, evaluates this integral?","kind":"action"},{"text":"What is the result, and can it be checked by differentiation?","kind":"measurement"}]}]},{"id":"numerical","name":"Numerical","description":"Numerical integration.","findings":[{"id":"numerical-finding","name":"Numerical","description":"Numerical integration.","questions":[{"text":"How can this integral be approximated numerically, and with what error?","kind":"action"},{"text":"When is numerical integration necessary?","kind":"boundary"}]}]}]},{"id":"meaning","name":"Meaning","description":"Interpretation.","layers":[{"id":"interpret","name":"Interpret","description":"What it represents.","findings":[{"id":"interpret-finding","name":"Interpret","description":"Interpretation.","questions":[{"text":"What does this integral represent physically or geometrically?","kind":"definition"},{"text":"What are the units of the result?","kind":"measurement"}]}]},{"id":"fundamental","name":"Fundamental","description":"Fundamental theorem.","findings":[{"id":"fundamental-finding","name":"Fundamental","description":"Fundamental theorem.","questions":[{"text":"How does the fundamental theorem of calculus connect integrals and derivatives?","kind":"definition"},{"text":"When does it apply?","kind":"boundary"}]}]}]},{"id":"advanced","name":"Advanced","description":"Generalisations.","layers":[{"id":"types","name":"Types","description":"Kinds of integral.","findings":[{"id":"types-finding","name":"Types","description":"Types.","questions":[{"text":"How do line, surface, multiple and Lebesgue integrals generalise the basic integral?","kind":"definition"},{"text":"When is each used?","kind":"provenance"}]}]},{"id":"improper","name":"Improper","description":"Improper integrals.","findings":[{"id":"improper-finding","name":"Improper","description":"Improper integrals.","questions":[{"text":"Does this improper integral converge, and what is a principal value?","kind":"boundary"},{"text":"Which tests apply?","kind":"definition"}]}]}]},{"id":"learn","name":"Learn","description":"Teaching.","layers":[{"id":"teach","name":"Teach","description":"Teaching integrals.","findings":[{"id":"teach-finding","name":"Teach","description":"Teaching.","questions":[{"text":"How can integration be taught with intuition and worked examples?","kind":"action"},{"text":"Which misconceptions arise?","kind":"provenance"}]}]},{"id":"applications","name":"Applications","description":"Uses in science.","findings":[{"id":"applications-finding","name":"Applications","description":"Applications.","questions":[{"text":"How are integrals used for quantities such as work, probability and optical path length?","kind":"provenance"},{"text":"Which examples help?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Compute and explain integrals, stating the method and conditions used.","Check that an integral converges before reporting its value."],"mustNot":["Report a divergent or improper integral as a finite value without stating so.","Hide numerical error in integrals used for dosages, loads or financial totals.","Do a student's graded work in a way that breaks their institution's rules."],"requiresHuman":["Using computed integrals in a safety-critical calculation that is signed off."]},"ethics":{"considerations":["Integrals in engineering, medicine and finance have real consequences when wrong.","Learners need help that builds understanding."],"affectedParties":["Students","Engineers and analysts","People affected by calculated results"]},"owners":{"steward":"Nobody owns the concept; it is held in common by mathematics.","roles":[],"masterSystems":[]},"relations":[{"target":"Q207643","type":"parent","note":"registry parent class"},{"target":"Q6498784","type":"parent","note":"registry parent class"},{"target":"Q657596","type":"parent","note":"registry parent class"},{"target":"unary operation","type":"related","note":"category"},{"target":"linear 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finite"]},"interfaces":{"applicability":"required","items":["ISO 80000-2 notation"]},"context":{"applicability":"required","items":["Calculus, analysis and quantitative science.","definite and indefinite integrals","multiple, line and surface integrals","Lebesgue and other measure-theoretic integrals","improper integrals and principal values","numerical integration"]}},"sources":[{"title":"Wikidata item Q80091: integral","url":"https://www.wikidata.org/wiki/Q80091","note":"identity and sense of the item"},{"title":"Wikipedia: Integral","url":"https://en.wikipedia.org/wiki/Integral","note":"general description of the item"}],"openQuestions":["Should each integral type be a separate entry?","How should computer algebra tools be linked?","How should applications be linked?"],"resources":{"spec":"/models/things/publications/thing-q80091/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q80091/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-11T19:59:16Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:32+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}