Math 2.0

Real functions Riemann and related integrals Theorem

Fundamental theorem of calculus

Builds on First fundamental theorem of calculusSecond fundamental theorem of calculusAntiderivative

Statement

Let a<ba < b and let f ⁣:[a,b]→ℝf \colon [a, b] \to \mathbb{R} be continuous. Then the function F(x)=∫axf(t) dtF(x) = \int_a^x f(t) \, dt is differentiable on (a,b)(a, b), with F′(x)=f(x)F'(x) = f(x). If GG is continuous on [a,b][a, b] and G′=fG' = f on (a,b)(a, b), then ∫abf(x) dx=G(b)−G(a).\int_a^b f(x) \, dx = G(b) - G(a).

Remarks

The first part is the first fundamental theorem. The second part is the second fundamental theorem. Together they say that differentiation and integration are inverse operations. The theorem reduces the computation of integrals to the search for antiderivatives.

History

James Gregory published the first statement and proof of a geometric form of the theorem. Isaac Barrow proved a more general version. Isaac Newton and Gottfried Leibniz then built the calculus around it.

Reading path

To understand this page from the ground up, read these 36 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionAntiderivative, definition, no evidence levelAntiderivativedefinitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementRiemann integral, definition, no evidence levelRiemann integraldefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementIntermediate value theorem, statement, no evidence levelIntermediate value th…statementDefinite integral, definition, no evidence levelDefinite integraldefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementAdditivity of the integral over intervals, statement, no evidence levelAdditivity of the int…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementMonotonicity of the integral, statement, no evidence levelMonotonicity of the i…statementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementMean value theorem for integrals, statement, no evidence levelMean value theorem fo…statementRolle's theorem, statement, no evidence levelRolle's theoremstatementFirst fundamental theorem of calculus, statement, no evidence levelFirst fundamental the…statementMean value theorem, statement, no evidence levelMean value theoremstatementSecond fundamental theorem of calculus, statement, no evidence levelSecond fundamental th…statementFundamental theorem of calculus, statement, no evidence levelFundamental theorem o…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Local maximum definition
  3. Partition of an interval definition
  4. Sequence definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Mesh of a partition definition
  8. Riemann sum definition
  9. Real sequence definition
  10. Subsequence definition
  11. Continuity at a point definition
  12. Derivative definition
  13. Riemann integrable function definition
  14. Bounded sequence definition
  15. Limit of a sequence definition
  16. Monotone sequence definition
  17. Continuous function definition
  18. Antiderivative definition
  19. Fermat's theorem on stationary points statement
  20. Riemann integral definition
  21. Convergent sequence definition
  22. Monotone convergence theorem for sequences statement
  23. Every sequence has a monotone subsequence statement
  24. Intermediate value theorem statement
  25. Definite integral definition
  26. Bolzano–Weierstrass theorem statement
  27. Additivity of the integral over intervals statement
  28. Boundedness theorem statement
  29. Monotonicity of the integral statement
  30. Extreme value theorem statement
  31. Mean value theorem for integrals statement
  32. Rolle's theorem statement
  33. First fundamental theorem of calculus statement
  34. Mean value theorem statement
  35. Second fundamental theorem of calculus statement
  36. Fundamental theorem of calculus statement

Where this leads

10 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus, Formal statements
About this page
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URIm2://0NHGY0N0JB
Kindstatement, theorem
WritingW1 stated
MSC26A42
StatusThis copy of the site has no build data. Run m2 status 0NHGY0N0JB.
UsesFirst fundamental theorem of calculus, Second fundamental theorem of calculus, Antiderivative
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