# integral over a period interval

Theorem.β If the real function $f$ is periodic and integrable (http://planetmath.org/RiemannIntegrable) over a period (http://planetmath.org/Periodic) interval, the value of integral over a period interval is always the same, i.e.

 $\displaystyle\int_{a}^{a+p}\!f(x)\,dx\;=\;\int_{0}^{p}f(x)\,dx\quad\forall\,a% \in\mathbb{R}$ (1)

where $p$ is the period of $f$.

Proof.β The right hand side of the equation (1) is manipulated, with one substitution (http://planetmath.org/ChangeOfVariableInDefiniteIntegral) β$x=t\!+\!p$:

 $\displaystyle\int_{0}^{p}f(x)\,dx$ $\displaystyle\;=\;\int_{0}^{a}f(x)\,dx+\int_{a}^{p}f(x)\,dx$ $\displaystyle\;=\;\int_{0}^{a}f(x)\,dx+\int_{a}^{a+p}\!f(x)\,dx-\int_{p}^{a+p}% \!f(x)\,dx$ $\displaystyle\;=\;\int_{0}^{a}f(x)\,dx+\int_{a}^{a+p}\!f(x)\,dx-\int_{0}^{a}f(% t\!+\!p)\,dt$ $\displaystyle\;=\;\int_{0}^{a}f(x)\,dx+\int_{a}^{a+p}\!f(x)\,dx-\int_{0}^{a}f(% t)\,dt$ $\displaystyle\;=\;\int_{a}^{a+p}\!f(x)\,dx$

## References

• 1 Ernst LindelΓΆf: Johdatus korkeampaan analyysiin. Fourth edition. Werner SΓΆderstrΓΆm OsakeyhtiΓΆ, Porvoo ja Helsinki (1956).
• 2 FrΓ₯ga Lund om matematik, http://www.maths.lth.se/query/here.
Title integral over a period interval IntegralOverAPeriodInterval 2013-03-22 18:43:57 2013-03-22 18:43:57 pahio (2872) pahio (2872) 7 pahio (2872) Theorem msc 26A15 msc 26A42 integral over a period integral of periodic function DefiniteIntegral IntegralsOfEvenAndOddFunctions