Let be continuous and periodic with period For any real numbers evaluate

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- May 22nd 2009, 08:55 AMNonCommAlgTechniques of integration (6)
Let be continuous and periodic with period For any real numbers evaluate

- May 22nd 2009, 11:54 AMTheAbstractionist
Hi

**NonCommAlg**.

A continuous function is integrable, and the integral over a bounded interval of an integrable periodic function is bounded. Hence, using the substitution

since the integral is bounded

Something tells me I may have done something wrong, because, well, surely it can’t be that simple … (Thinking) - May 22nd 2009, 02:12 PMOpalg
I'd agree with that approach up to that point, but I wouldn't go on to say that the limit is 0, because the length of the interval (in the u-integral) is getting unboundedly long.

Let be the mean value of f over one period. Then the interval [na,nb] consists of subintervals of length T (not counting odd bits at the ends, which we can dispose of with epsilons). So . That's my candidate for the limit. - May 22nd 2009, 03:15 PMTheAbstractionist
- May 22nd 2009, 06:54 PMNonCommAlg