Math 2.0

Real functions Theorem

Linearity 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, and let α,β∈ℝ\alpha, \beta \in \mathbb{R}. Then αf+βg\alpha f + \beta g is Riemann integrable on [a,b][a, b], and ∫ab(αf(x)+βg(x)) dx=α∫abf(x) dx+β∫abg(x) dx.\int_a^b (\alpha f(x) + \beta g(x)) \, dx = \alpha \int_a^b f(x) \, dx + \beta \int_a^b g(x) \, dx.

Remarks

The Riemann sums satisfy S(αf+βg,P,t)=αS(f,P,t)+βS(g,P,t)S(\alpha f + \beta g, P, t) = \alpha S(f, P, t) + \beta S(g, P, t). Pass to the limit. So the Riemann integrable functions on [a,b][a, b] form a vector space, and the integral is a linear map on it. Mathlib states additivity as intervalIntegral.integral_add.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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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 integraldefinitionLinearity of the integral, statement, no evidence levelLinearity of the inte…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. Linearity of the integral statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Integral, Linearity
About this page
Tag080RWVVSB6
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Kindstatement, theorem
WritingW0 stub
MSC26A42
StatusThis copy of the site has no build data. Run m2 status 080RWVVSB6.
UsesDefinite integral, Riemann sum

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