Math 2.0

statement theorem

Linearity of the integral

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.

Explanations

L1 human

The integral of a sum is the sum of the integrals. Constant factors can move out of the integral.

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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
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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

Sources

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

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