PlanetMath: integral over a period interval

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integral over a period interval

(Theorem)

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.

$\displaystyle \int_a^{a+p}\!f(x)\,dx \;=\; \int_0^pf(x)\,dx \quad \forall\,a \in \mathbb{R}$

(1)

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

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

$\displaystyle \int_0^pf(x)\,dx$	$\displaystyle \;=\; \int_0^af(x)\,dx+\int_a^pf(x)\,dx$
	$\displaystyle \;=\; \int_0^af(x)\,dx+\int_a^{a+p}\!f(x)\,dx-\int_p^{a+p}\!f(x)\,dx$
	$\displaystyle \;=\; \int_0^af(x)\,dx+\int_a^{a+p}\!f(x)\,dx-\int_0^{a}f(t\!+\!p)\,dt$
	$\displaystyle \;=\; \int_0^af(x)\,dx+\int_a^{a+p}\!f(x)\,dx-\int_0^{a}f(t)\,dt$
	$\displaystyle \;=\; \int_a^{a+p}\!f(x)\,dx$

Bibliography

1: ERNST LINDELÖF: Johdatus korkeampaan analyysiin. Fourth edition. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
2: Fråga Lund om matematik, here.

"integral over a period interval" is owned by pahio.

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Other names:

integral over a period, integral of periodic function

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Cross-references: equation, right hand side, proof, interval, periodic, real function, theorem

This is version 4 of integral over a period interval, born on 2009-01-13, modified 2009-01-14.
Object id is 11501, canonical name is IntegralOverAPeriodInterval.
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Classification:

AMS MSC:	26A15 (Real functions :: Functions of one variable :: Continuity and related questions )
	26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type)

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