Math 2.0

statement theorem

Triangle inequality for integrals

Statement

Let f ⁣:[a,b]→Rf \colon [a, b] \to \mathbb{R} be Riemann integrable. Then ∣f∣|f| is Riemann integrable on [a,b][a, b], and ∣∫abf(x) dx∣≤∫ab∣f(x)∣ dx.\Big| \int_a^b f(x) \, dx \Big| \le \int_a^b |f(x)| \, dx.

Remarks

The inequality follows from −∣f∣≤f≤∣f∣-|f| \le f \le |f| and the monotonicity of the integral. The converse of the first claim fails. On [0,1][0, 1], let f=1f = 1 on the rationals and f=−1f = -1 elsewhere. Then ∣f∣=1|f| = 1 is integrable, but ff is not. Mathlib states the inequality as intervalIntegral.norm_integral_le_integral_norm.

Explanations

L1 human

Cancellation can only make an integral smaller in size. The size of the integral is at most the integral of the size of the function.

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To read this node, read these 16 nodes in this order. The foundations come first, and the list ends with this node.

Partition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionRiemann integral, definition, no evidence levelRiemann integraldefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionDefinite integral, definition, no evidence levelDefinite integraldefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionMonotonicity of the integral, statement, no evidence levelMonotonicity of the i…statementAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionTriangle inequality for integrals, statement, no evidence levelTriangle inequality f…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. Partition of an interval definition
  2. Sequence definition
  3. Mesh of a partition definition
  4. Riemann sum definition
  5. Real sequence definition
  6. Riemann integrable function definition
  7. Limit of a sequence definition
  8. Series definition
  9. Riemann integral definition
  10. Convergent sequence definition
  11. Partial sum definition
  12. Definite integral definition
  13. Convergent series definition
  14. Monotonicity of the integral statement
  15. Absolute convergence definition
  16. Triangle inequality for integrals statement

Sources

  1. Web https://en.wikipedia.org/wiki/Integral, Inequalities

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