Average Value Function and Equivalent Periodic Function

**mathisfunforme** · February 13th 2010, 06:12 PM

This is a homework problem and I'm not even sure where to start with it. The problem reads: "Show that if f(x) has period p, the average value of f is the same over any interval of length p. Hint: $\int_a^{a+p}f(x) dx$ as the sum of two integrals (a to p, and p to a+p) and make the change of variable x = t+p in the second integral."

Okay so I know how to write the hint out and I get the point that f(t+p) = f(t) but what am I actually supposed to write out? I don't know out to transform this integral. Where do I start after writing out the 'hint'?

**HallsofIvy** · February 14th 2010, 03:31 AM

So you have gotten to
$\int_a^{a+p} f(x) dx= \int_a^p f(x)dx+ \int_p^{a+p} f(x)dx$
and then let x= t+ p (or t= x- p) to get
$\int_a^p f(x)dx+ \int_0^a f(t+p)dt= \int_a^p f(x)dx+ \int_0^a f(t)dt$ .

But now, since the "x" and "t" in the integrals are "dummy variables" (they don't appear in the final integral so we can change letters at will), we can write that as
$\int_a^p f(x)dx+ \int_0^a f(x)dx$

Now, put those integrals back together into one integral.

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