Find the average of $\sin^{100} (x)$ in 5 minutes?

Question

I read this quote attributed to VI Arnold.

"Who can't calculate the average value of the one hundredth power of the sine function within five minutes, doesn't understand mathematics - even if he studied supermanifolds, non-standard calculus or embedding theorems."

EDIT Source is "A mathematical trivium" A book of 100 problems that university students "should be able to solve". The statement asks for calculation within 10% accuracy.

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So the average value over the entire domain should be the same as the average value over $[0,\pi/2]$

$$\langle\sin^{100} (x)\rangle= \frac{\int_0^{\pi/2} \sin^{100}(x) dx}{\int_0^{\pi/2} dx}.$$

So here's what I did:

First, this graph would be a train of highly sharp peaks. The integrand would assume values close to zero a up till before it sharply rises to 1.

So up till some $\epsilon \in [0,\pi/2]$ we will have $\sin x \approx x$ and for the remaining $\pi/2 - \epsilon$ interval I could find the area of triangle with base $\pi/2 - \epsilon$ and height $1$

$$\langle \sin^{100} (x)\rangle \approx \frac{2}{\pi} \left(\int_0^\epsilon x^{100} dx + .5 (\frac{\pi}{2}-\epsilon)\right).$$

I believe in principal it should be possible to find an $\epsilon$ such that the above expression yields the exact answer. So I try to approximate it, no good. Then I try mathematica and it is looking like there is no $\epsilon$ for which the value I am expecting is even close to the actual value. I plot the original and find that my approximation is hopeless.

Not to mention that my 5 minutes were over. So I admit I do not understand mathematics and humbly ask if someone could:

1. Point out my mistake (Other than that $\epsilon$ is probably incomputable within 5 mins)
2. How the hell is this done in 5 minutes?

The picture below has the $\sin^{100} x$ in blue (bottom) and my approximation of it plotted against $\epsilon$ (pink). Although there is no reason for them to be together, the upper graph has a minima quite above the exact value of the integral.

EDIT Just realized

Let $$u=\cos x.$$

$$\int_0^{\pi/2} \sin^{100}(x) dx = \int_0^1 (1-u^2)^{99/2}du\approx \int_0^1 \left(1 - \frac{99}{2} u^2\right) du $$

Answer 1

Since $$\sin x=\frac{e^{ix}-e^{-ix}}{2i} $$ and, for $k\in\mathbb Z $, $$\int_0^{2\pi}e^{ikx}\,dx=\left\{\begin{array}{cl}0&k\ne0\\ 2\pi&k=0\end{array}\right.,$$ we have $$\int_0^{2\pi}\sin^{100}x\,dx=\frac1{2^{100}}\sum_{k=0}^{100}\binom{100}{k}\int_0^{2\pi}e^{ikx}(-1)^{100-k}e^{-i(100-k)x}\,dx=\frac{\binom{100}{50}}{2^{100}}2\pi,$$ and the average value is $$\frac{\binom{100}{50}}{2^{100}}.$$

Answer 2

If you do an integration by parts, writing $\sin^{100}x$ as $\sin(x)\sin^{99}(x)$ you get:

$$\int_{-a}^a \sin^{100}x dx = \frac{99}{100} \int_{-a}^a \sin^{98}x dx$$

So I think that argument gives you an induction, and you get something like:

$$\int_{-\pi}^{\pi} \sin^{100}x dx = \frac{99\cdot 97 \cdots 1}{100 \cdot 98 \cdots 2} 2\pi$$

FYI: I've never studied supermanifolds. I am a professional mathematician and I think the problem took me maybe 2 minutes, once I had finished reading your question.

FYI number 2: A decent approximation to the average of $\sin^n(x)$ over $[-\pi,\pi]$ would be $\frac{1}{\sqrt{n}}$. You get this from the above argument, using the approximation that $\ln(1+x) \simeq x$ for $x$ small, together with the approximation that the sum $1+1/2+1/3+\cdots+1/n \simeq \ln(n)$.

Answer 3

According to a computer algebra package I'm using, which are pretty powerful these days, $\int_{0}^{\pi/2} \frac{2\sin^{100}(x)}{\pi} dx = \frac{126114180XXXXXX24166851562157}{158456325028528675187087900672}$ (the point here being that even if one is not so great on methods of integration, computer algebra systems have come quite far in the past few decades). It's also worth pointing out that this answer is conceptually much less satisfying than either Andres or Ryan's answer. (six of the characters of the answer have been replaced by "X"s, because I don't want to just give you the answer).

Moreover, there's the question of the intent of VI Arnold's quote, which is difficult to understand without his full biography. If you could perhaps provide a reference to the quote so the context is available, then it may be possible to explicate it, but that's kind of soft-questiony.

Answer 4

Assuming Arnold means to find an approximate value -- I'd do this as follows: first, we may as well find the average of $\cos^{100} x$. I'll do this over a half-period, $-\pi/2 \le x \le \pi/2$. But $\cos x \approx 1-x^2/2$, so $\cos^{100} x \approx (1-x^2/2)^{100}$. If $x$ is small -- which it will have to be for $\cos x$ to be large (i. e. near 1) -- then $1-x^2/2 \approx e^{-x^2/2}$. So $\cos^{100} x \approx e^{-50x^2}$.

So the number we're looking for is about $$ {1 \over \pi} \int_{-\pi/2}^{\pi/2} e^{-50x^2} \: dx. $$ But the integrand is so small far from zero that the limits of integration can be replaced with $-\infty$ and $\infty$ without changing much. That gives $$ {1 \over \pi} \int_{-\infty}^\infty e^{-50x^2} \: dx. $$ Change variables, $u = x/\sqrt{50}$, to get $$ {1 \over \pi \sqrt{50}} \int_{-\infty}^\infty e^{-u^2} \: du. $$ Finally, that integral is well-known to be $\sqrt{\pi}$; the approximate answer is $1/\sqrt{50\pi}$.

This method has the advantage that it works for high powers of any function and isn't specialized to the trig functions. One source that explains this trick (and uses it to approximate the same integral) is Sanjoy Mahajan's book Street-Fighting Mathematics (link goes to Creative-Commons downloadable version of the book).

Answer 5

I suspect Michael Lugo's answer was the intended one, but for what it's worth, Andres Caicedo's combinatorial answer has a combinatorial proof. The Riemann sum

$$\frac{1}{n} \sum_{k=0}^{n-1} \cos^{100} \frac{2 \pi k}{n}$$

counts the probability that you return to where you started in a random walk of length $100$ on $\mathbb{Z}/n\mathbb{Z}$ (where adjacent residues are connected by an edge); this is because the adjacency matrix is $P + P^{-1}$ where $P$ is a permutation matrix describing an $n$-cycle, so the eigenvalues of the adjacency matrix are $e^{ \frac{2 \pi i k}{n} } + e^{- \frac{2 \pi i k}{n} } = 2 \cos \frac{2 \pi k}{n}$.

For $n > 100$ this probability is clearly just $\frac{1}{2^{100}} {100 \choose 50}$, and taking the limit as $n \to \infty$ we obtain our result. And combinatorialists know that ${2n \choose n} \approx \frac{4^n}{\sqrt{\pi n}}$ by one of various methods (Stirling's formula, singularity analysis, the central limit theorem...).

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Actually (I had forgotten that I knew this) one can get the integral directly without looking at a Riemann sum. The key is that $\mathbb{Z}$ (the graph where adjacent integers are connected by an edge) is the representation graph of $\text{SO}(2)$ acting on its standard representation $V$, so

$$\int_0^{2\pi} (2 \cos x)^{100} \, dx$$

is precisely the multiplicity of the trivial representation in $V^{\otimes 100}$, which is precisely the number of walks of length $100$ from the origin to itself on $\mathbb{Z}$.