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单词 IntegralOverAPeriodInterval
释义

integral over a period interval


Theorem.  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.

aa+pf(x)𝑑x=0pf(x)𝑑xa(1)

where p is the period of f.

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

0pf(x)𝑑x=0af(x)𝑑x+apf(x)𝑑x
=0af(x)𝑑x+aa+pf(x)𝑑x-pa+pf(x)𝑑x
=0af(x)𝑑x+aa+pf(x)𝑑x-0af(t+p)𝑑t
=0af(x)𝑑x+aa+pf(x)𝑑x-0af(t)𝑑t
=aa+pf(x)𝑑x

References

  • 1 Ernst Lindelöf: Johdatus korkeampaan analyysiin. Fourth edition. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
  • 2 Fråga Lund om matematik, http://www.maths.lth.se/query/here.
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更新时间:2025/5/4 20:14:28