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I am stuck on a question that involves the intergral of a periodic function. The question is phrased as follows:

A function is periodic with period $a$, if $f(x)=f(x+a)$ for all $x$.

Question

If $f$ is continuous and periodic with period $a$, then show that $$\int_{0}^{a}f(t)dt=\int_{b}^{b+a}f(t)dt$$ for all $b\in \mathbb{R}$.

I understand the equality, but I am having trouble showing that it is true for all $b$. I've tried writing it in different forms such as $F(a)=F(b+a)-F(b)$. This led me to the following, though I am not sure how this shows the equality is true for all $b$,

$$\int_{0}^{a}f(t)dt-\int_{b}^{b+a}f(t)dt=0$$ $$=F(a)-F(0)-F(b+a)-F(b)$$ $$=(F(b+a)-F(a))-F(b)$$ $$=\int_{a}^{b+a}f(t)dt-\int_{0}^{b+a}f(t)dt=0$$

So, this leaves me with

$$\int_{a}^{b+a}f(t)dt-\int_{0}^{b+a}f(t)dt=\int_{0}^{a}f(t)dt-\int_{b}^{b+a}f(t)dt$$

I feel I am close, and I've made myself a diagram of a sine function to visualize what each of the above integrals might describe, but the power to explain the above equality evades me.

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See here for some proofs. –  t.b. Jan 12 '12 at 8:09
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4 Answers 4

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Let $H(x)=\int_x^{x+a}f(t)\,dt$. Then $$\frac{dH}{dx}=f(x+a)-f(x)=0.$$ It follows that $H(x)$ is constant. In particular, $H(b)=H(0)$.

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You have made various false steps in your four line block and should have ended up with $$\int_{a}^{b+a}f(t)dt-\int_{0}^{b}f(t)dt=0$$ but this does not take you much further forward.

Instead note that somewhere in the interval $[b, b+a]$ is an integer multiple of $a$, say $na$. Then using $f(t)=f(t+a)=f(t+na)$: $$\int_{b}^{b+a}f(t)dt = \int_{b}^{na}f(t)dt+\int_{na}^{b+a}f(t)dt = \int_{b+a}^{(n+1)a}f(t)dt+\int_{an}^{b+a}f(t)dt = \int_{na}^{(n+1)a}f(t)dt = \int_{0}^{a}f(t)dt.$$

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With no differentiation: pick the unique integer $n$ such that $b\leqslant na\lt b+a$, decompose the integral of $f(t)$ over $t$ from $b$ to $b+a$ into the sum of the integrals from $b$ to $na$ and from $na$ to $b+a$, apply the changes of variable $t=x+(n-1)a$ in the former and $t=x+na$ in the latter, then the periodicity of $f$ implies that $f(x)=f(t)$, hence the result is the sum of the integrals of $f(x)$ over $x$ from $b-(n-1)a$ to $a$ and from $0$ to $b-(n-1)a$... et voilà !

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We have $$ \int_{0}^{a}f(t)\ dt+\int_{a}^{a+b}f(x)\ dx=\int_{0}^{b}f(y)\ dy+\int_{b}^{a+b}f(t)\ dt, $$ and setting $x=y-a$ turns the second integral into the third one.

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