Math 2.0

statement theorem

Darboux and Riemann integrals agree

Statement

Let a<ba < b and let f ⁣:[a,b]→Rf \colon [a, b] \to \mathbb{R} be bounded. Then ff is Riemann integrable if and only if its upper and lower Darboux integrals are equal. In this case the common value is the Riemann integral ∫abf(x) dx\int_a^b f(x)\, dx.

Remarks

Each Riemann sum lies between the lower and upper Darboux sums. This gives one direction. For the other direction, one shows that Darboux sums for partitions of small mesh are close to the Darboux integrals. So the Riemann and the Darboux approaches define the same integral.

Explanations

L1 human

There are two standard ways to define the integral, with Riemann sums or with upper and lower sums. They give the same integrable functions and the same values.

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To read this node, read these 11 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…definitionBounded sequence, definition, no evidence levelBounded sequencedefinitionRiemann integral, definition, no evidence levelRiemann integraldefinitionUpper and lower Darboux sums, definition, no evidence levelUpper and lower Darbo…definitionUpper and lower Darboux integrals, definition, no evidence levelUpper and lower Darbo…definitionDarboux and Riemann integrals agree, statement, no evidence levelDarboux and Riemann i…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. Bounded sequence definition
  8. Riemann integral definition
  9. Upper and lower Darboux sums definition
  10. Upper and lower Darboux integrals definition
  11. Darboux and Riemann integrals agree statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Darboux_integral, Refinement of a partition and relation to Riemann integration

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