integral over a period interval

Theorem.  If the real function f is periodic and integrable ( over a period ( interval, the value of integral over a period interval is always the same, i.e.

∫aa+pf⁒(x)⁒𝑑x=∫0pf⁒(x)⁒𝑑xβ€ƒβˆ€aβˆˆβ„ (1)

where p is the period of f.

Proof.  The right hand side of the equation (1) is manipulated, with one substitution (  x=t+p:

∫0pf⁒(x)⁒𝑑x  =∫0af⁒(x)⁒𝑑x+∫apf⁒(x)⁒𝑑x


  • 1 Ernst LindelΓΆf: Johdatus korkeampaan analyysiin. Fourth edition. Werner SΓΆderstrΓΆm OsakeyhtiΓΆ, Porvoo ja Helsinki (1956).
  • 2 FrΓ₯ga Lund om matematik,
Title integral over a period interval
Canonical name IntegralOverAPeriodInterval
Date of creation 2013-03-22 18:43:57
Last modified on 2013-03-22 18:43:57
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Theorem
Classification msc 26A15
Classification msc 26A42
Synonym integral over a period
Synonym integral of periodic function
Related topic DefiniteIntegral
Related topic IntegralsOfEvenAndOddFunctions