Math 2.0

Real functions Theorem

Second fundamental theorem of calculus

Builds on Mean value theoremRiemann sumRiemann integrable function

Statement

Let a<ba < b, and let F ⁣:[a,b]→ℝF \colon [a, b] \to \mathbb{R} be continuous on [a,b][a, b] and differentiable on (a,b)(a, b). Assume that F′F' extends to a Riemann integrable function ff on [a,b][a, b]. Then ∫abf(x) dx=F(b)−F(a).\int_a^b f(x) \, dx = F(b) - F(a).

Remarks

For a partition PP, the mean value theorem gives tags tit_i with F(xi)−F(xi−1)=f(ti)(xi−xi−1)F(x_i) - F(x_{i-1}) = f(t_i)(x_i - x_{i-1}). So F(b)−F(a)F(b) - F(a) is a Riemann sum of ff for each partition. One writes F(b)−F(a)=[F(x)]abF(b) - F(a) = \big[ F(x) \big]_a^b. The hypothesis that ff is integrable is needed. Some differentiable functions have derivatives that are not Riemann integrable. Mathlib states it as intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le.

Reading path

To understand this page from the ground up, read these 27 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 functiondefinitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementConvergent 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 …statementBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementRolle's theorem, statement, no evidence levelRolle's theoremstatementMean value theorem, statement, no evidence levelMean value theoremstatementSecond fundamental theorem of calculus, statement, no evidence levelSecond fundamental th…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. Fermat's theorem on stationary points statement
  19. Convergent sequence definition
  20. Monotone convergence theorem for sequences statement
  21. Every sequence has a monotone subsequence statement
  22. Bolzano–Weierstrass theorem statement
  23. Boundedness theorem statement
  24. Extreme value theorem statement
  25. Rolle's theorem statement
  26. Mean value theorem statement
  27. Second fundamental theorem of calculus statement

Where this leads

39 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus, Second part
About this page
Tag0XYAFCN0PT
URIm2://0XYAFCN0PT
Kindstatement, theorem
WritingW1 stated
MSC26A42
StatusThis copy of the site has no build data. Run m2 status 0XYAFCN0PT.
UsesMean value theorem, Riemann sum, Riemann integrable function

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