Math 2.0

statement theorem

Mean value theorem for integrals

Statement

Let a<ba < b and let f ⁣:[a,b]→Rf \colon [a, b] \to \mathbb{R} be continuous. Then there is c∈[a,b]c \in [a, b] with ∫abf(x) dx=f(c) (b−a).\int_a^b f(x) \, dx = f(c) \, (b - a).

Remarks

Let mm and MM be the minimum and the maximum of ff, from the extreme value theorem. By monotonicity, the average 1b−a∫abf\frac{1}{b - a} \int_a^b f lies in [m,M][m, M]. The intermediate value theorem gives cc. The number 1b−a∫abf\frac{1}{b - a} \int_a^b f is the average value of ff on [a,b][a, b]. Mathlib has a form for averages, exists_eq_interval_average.

Explanations

L1 human

A continuous function takes its average value somewhere. The area under the graph equals the area of a rectangle whose height is a value of ff.

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To read this node, read these 25 nodes in this order. The foundations come first, and the list ends with this node.

Limit point of a set, definition, no evidence levelLimit point of a setdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionMesh 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 pointdefinitionRiemann 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 functiondefinitionRiemann 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…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…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. Partition of an interval definition
  3. Sequence definition
  4. Limit of a function definition
  5. Mesh of a partition definition
  6. Riemann sum definition
  7. Real sequence definition
  8. Subsequence definition
  9. Continuity at a point definition
  10. Riemann integrable function definition
  11. Bounded sequence definition
  12. Limit of a sequence definition
  13. Monotone sequence definition
  14. Continuous function definition
  15. Riemann integral definition
  16. Convergent sequence definition
  17. Monotone convergence theorem for sequences statement
  18. Every sequence has a monotone subsequence statement
  19. Intermediate value theorem statement
  20. Definite integral definition
  21. Bolzano–Weierstrass theorem statement
  22. Boundedness theorem statement
  23. Monotonicity of the integral statement
  24. Extreme value theorem statement
  25. Mean value theorem for integrals statement

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8 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Mean_value_theorem, First mean value theorem for definite integrals

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