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Could someone give a suggestion to calculate this limit please?

$ \lim_{a \to \infty} \int\limits_0^{2010} \sqrt{x(1+\cos ax)}\ \mathrm dx $

I don't know methods for solving such limits with integrals. Thanks.

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    Was this problem from$a$contest? The "$2010$" makes me think that maybe it was. I'm just curious.2011-09-14

1 Answers 1

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This is one of those cases where you don't need a general method -- you need a trick (by which I mean a method that you don't expect to generalize to other problems).

The crucial idea is that when $a$ goes towards infinity, you can average over the cosine wave everywhere. Making this work rigorously is a bit involved, though:

For notational convenience, set $M=2010$ and $\nu = \frac{a}{2\pi}$, and let $L(\nu)=\int_0^M \sqrt{x(1+\cos 2\pi\nu x)}\,dx$. We're then seeking $\lim_{\nu\to\infty}L(\nu)$. It is clear that $\int_0^M \sqrt{\frac{\lfloor \nu x\rfloor}{\nu}}\sqrt{1+\cos 2\pi\nu x}\,dx \le L(\nu) \le \int_0^M \sqrt{\frac{\lceil\nu x\rceil}{\nu}}\sqrt{1+\cos 2\pi\nu x}\,dx$ If for a moment we restrict our attention to $\nu$s that are multiples of $\frac{1}M$, we can integrate each wavelength (where $\lfloor \nu x \rfloor$ resp. $\lceil \nu x \rceil$ are constant) separately and combine to find $K \int_0^M \sqrt{\frac{\lfloor \nu x\rfloor}{\nu}}\,dx \le L(\nu) \le K \int_0^M \sqrt{\frac{\lceil\nu x\rceil}{\nu}}\,dx$ where $K = \int_0^1 \sqrt{1+\cos 2\pi u}\,du = \frac{1}{2\pi} \int_0^{2\pi} \sqrt{1+\cos t}\,dt$

However, the integrals surrounding $L(\nu)$ are now just Riemann sums for $\int_0^M \sqrt{x}\,dx$, and so both bounds for $L(\nu)$ approach $K\int_0^M\sqrt{x}\,dx = \frac{1}{2\pi} \int_0^{2\pi} \sqrt{1+\cos t}\,dt \int_0^M\sqrt{x}\,dx = \frac{4\sqrt 2}{2\pi}\cdot\frac{2}{3}\cdot M^{3/2} $ which is therefore the only possible limit.

It remains only to see that the $\nu$s that are not multiples of $\frac{1}{M}$ cannot prevent the limit from existing. But even if the average over the last partial wavelength next to $x=M$ differs from $K$, the net error resulting from this can be at most $\frac 2\nu$, which goes towards zero too.

(After having done all this, we can see that the trick actually is a general method for finding $\lim_{\nu\to\infty} \int_a^b f(x)g(\nu x)\,dx$ when $g$ is a periodic function. But how often do you run into such problems?)

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    Uh... yeah, you beat *me* to it, also... !2011-09-14