# Integral

**Type:** Concept  
**Domain:** Analysis  
**Codex URL:** /mathematics/analysis/integral/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Computes total accumulation, most famously area under a curve. Defined as a limit of Riemann sums.

## Fundamental Theorem of Calculus
Part 1: F(x)=∫ₐˣf(t)dt ⟹ F'(x)=f(x)
Part 2: ∫ₐᵇf(x)dx = F(b)−F(a), F any antiderivative of f

## Lebesgue integration
Lebesgue (1902 thesis) partitions the range instead of domain, using measure theory. Handles functions (like the Dirichlet function) that Riemann integration cannot.

## Historical notation
Leibniz introduced ∫ (elongated "S" for summa) in 1675.

## Sources
### Tier 1
- Rudin, W. (1976). *Principles of Mathematical Analysis*. 3rd ed.
- Royden, H.L. and Fitzpatrick, P. (2010). *Real Analysis*. 4th ed. Pearson.
### Tier 2
- Spivak, M. (2008). *Calculus*. 4th ed.
### Tier 3
- Boyer, C.B. (1949). *The History of the Calculus*. Dover.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*