Math 2.0

statement theorem

Second fundamental theorem of calculus

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.

Explanations

L1 human

To integrate ff, find a function FF with derivative ff. The integral is the change of FF from aa to bb.

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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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  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

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Sources

  1. Web https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus, Second part

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