Math 2.0

Real functions Theorem

Monotonicity of the integral

Builds on Definite integralRiemann sum

Statement

Let f,g ⁣:[a,b]→ℝf, g \colon [a, b] \to \mathbb{R} be Riemann integrable with f(x)≤g(x)f(x) \le g(x) for all x∈[a,b]x \in [a, b]. Then ∫abf(x) dx≤∫abg(x) dx.\int_a^b f(x) \, dx \le \int_a^b g(x) \, dx.

Remarks

Each Riemann sum of ff is at most the Riemann sum of gg for the same tagged partition. In particular, if m≤f≤Mm \le f \le M on [a,b][a, b], then m(b−a)≤∫abf≤M(b−a)m (b - a) \le \int_a^b f \le M (b - a). Mathlib states it as intervalIntegral.integral_mono_on.

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Partition of an interval, definition, no evidence levelPartition of an inter…definitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionRiemann integral, definition, no evidence levelRiemann integraldefinitionDefinite integral, definition, no evidence levelDefinite integraldefinitionMonotonicity of the integral, statement, no evidence levelMonotonicity of the i…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.
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  1. Partition of an interval definition
  2. Mesh of a partition definition
  3. Riemann sum definition
  4. Riemann integrable function definition
  5. Riemann integral definition
  6. Definite integral definition
  7. Monotonicity of the integral statement

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54 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Integral, Inequalities
About this page
Tag0MBWQKM8QR
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Kindstatement, theorem
WritingW0 stub
MSC26A42
StatusThis copy of the site has no build data. Run m2 status 0MBWQKM8QR.
UsesDefinite integral, Riemann sum

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