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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.


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 $1$

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

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?enter image description here

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 $$u=\cos x.$$

$$\int_0^{\pi/2} \sin^{100}(x) dx = \int_1^0 (1-u^2)^{49/2}du\approx \int_1^0 \left(1 - \frac{49}{2} u^2\right) du $$

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You could use integration by parts to calculate $\int\sin^n(x)\,dx$ in terms of $\int\sin^{n-2}(x)\,dx$ and use induction on $n$. –  George Lowther Mar 2 '11 at 2:35
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Your approximation using a triangle does not sound good. Better to write (shifting the peak to 0) $\cos x\approx 1-x^2/2\approx e^{-x^2/2}$, so $\cos^{100}(x)\approx e^{-50 x^2}$, which has integral $\sqrt{\pi/50}$. But, as I mentioned, you can also find an exact expression using integration by parts. Comparing the two methods should also give you a product expansion for $\pi$. –  George Lowther Mar 2 '11 at 2:42
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...and the approximation in your edit will not be accurate. $(1-u^2)^{\frac{n}{2}}$ will be approximated well by $e^{-\frac{n}{2}u^2}$, not by $1-\frac{n}{2}u^2$. –  George Lowther Mar 2 '11 at 2:55
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math.stackexchange.com/questions/20397/… –  user17762 Mar 2 '11 at 2:59
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Is no one going to mention that Arnold's quote is presumably meant to be at least somewhat provocative and is, in any case, far from being universally true? Rather than divulge whether or not I could meet this challenge (well, okay: I certainly didn't meet it; I had no interest in spending even one second trying to calculate this integral) let me say that by all accounts Alexander Grothendieck, for instance, would have trouble with it, and if he didn't/doesn't understand mathematics, there is not much hope for the rest of us. –  Pete L. Clark Mar 2 '11 at 11:23

9 Answers 9

up vote 88 down vote accepted

Since $$\sin x=\frac{e^{ix}-e^{-ix}}{2i} $$ and $$\int_0^{2\pi}e^{ikx}\,dx=\left\{\begin{array}{cl}0&k\ne0\\ 2\pi&k=0\end{array}\right.,$$ 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 $$\frac{\binom{100}{50}}{2^{100}}.$$

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+1. Looking at the way you have written the final answer, it begs for a combinatorial/probabilistic argument for the integral. Unfortunately, I am unable to think of a combinatorial/probabilistic immediately. –  user17762 Mar 2 '11 at 21:06
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@Sivaram: did you see my answer? –  Qiaochu Yuan Mar 18 '11 at 11:57
    
+1. Very slick. –  Isaac Solomon Jul 24 '13 at 4:00
    
By Maple: $$evalf(2*(Int(sin(x)^{100}, x = 0 .. (1/2)*Pi))/Pi, 20) $$ $$0.079589237387178761494. $$ –  user64494 Jun 9 at 12:32

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...).


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 $\mathbb{Z}$.

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I didn't see your combinatorial answer before. Nice one. –  user17762 Mar 20 '11 at 2:45

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).

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If instead of the 100th power we had any even power $n$, this method would give $1/\sqrt{n\pi}$. Andres Caicedo's method for finding the exact answer would give {n \choose n/2}/2^n. Applying Stirling's approximation for the factorial to Andres' answer gives my approximate answer. –  Michael Lugo Mar 2 '11 at 5:13
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For what it's worth: Andres's [exact answer](wolframalpha.com/input/?i=(100+choose+50)/(2%5E(100)%29) is about 0.079589… and this approximation is about 0.079788…, reasonably close. –  ShreevatsaR Mar 2 '11 at 7:12

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 $1+1/2+1/3+\cdots+1/n \simeq \ln(n)$.

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Actually, to leading order, it is approximated by $\sqrt{\frac{2}{n\pi}}$. So you seem to be out by a constant factor in the last paragraph. –  George Lowther Mar 2 '11 at 3:19
    
I don't know what "leading order" means. You can see what approximations I'm using so it's certainly not meant to be terribly good. –  Ryan Budney Mar 2 '11 at 4:03
    
You can give an asymptotic expansion in powers of $n$, for which $\sqrt{\frac{2}{n\pi}}$ is the dominant term for large $n$, which is what I mean by the "leading order term". So, the average is asymtotically equal to $\sqrt{\frac{2}{n\pi}}$. –  George Lowther Mar 2 '11 at 20:28

Note that $$\int_{0}^{\pi/2}\sin^{100}(x) \ dx=\frac{1}{2}\beta\left(\frac{101}{2},\frac{1}{2}\right)=\frac{\Gamma\left(\frac{101}{2}\right)\Gamma\left(\frac{1}{2}\right)}{2\Gamma(51)}=\frac{\binom{100}{50}}{2^{100}}$$ (You can take more than five minutes to write the answer here in Latex, but surely, it will take less than 2 minutes with a calculator)

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$\newcommand{\+}{^{\dagger}} \newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\down}{\downarrow} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\isdiv}{\,\left.\right\vert\,} \newcommand{\ket}[1]{\left\vert #1\right\rangle} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert} \newcommand{\wt}[1]{\widetilde{#1}}$ \begin{align} \color{#00f}{\large\int_{0}^{2\pi}\sin^{100}\pars{x}\,{\dd x \over 2\pi}}&= \oint_{\verts{z}\ = \ 1}{\pars{z^{2} - 1 \over 2\ic z}}^{100} \,{\dd z/\pars{\ic z} \over 2\pi} ={1 \over 2^{100}}\,{\rm Res}\bracks{% {\pars{1 - z^{2}}^{100} \over z^{\color{#c00000}{\large 100} + 1}}}_{\,z\ =\ 0} \end{align}

The residue 'arises' from the power $\ds{z^{100} = \pars{-z^{2}}^{\color{#c00000}{\large 50}}}$ in the numerator binomial expansion. Result: $\color{#00f}{\large\ds{{1 \over 2^{100}}\,{100 \choose 50}}}$.

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Average value of $\sin^{2n}(x)$ on $[0,\pi/2]$ is the coefficient of $x^n$ in the series expansion of $(1-x)^{-\frac{1}{2}}$, which is $\binom {2n}{n}2^{-2n}$. This is asypmtotically $1/\sqrt{n\pi}$.

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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.

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Added the source in an edit. –  Please Delete Account Mar 2 '11 at 16:07

OK, I'm going to go out on a limb with a naïve answer because I don't understand what's wrong with it. In that sense this is also a question - is there anything wrong with my analysis? Please be gentle if I am making a stupid mistake -- it just doesn't seem as complicated as all these answers appear.

In order to be "within 10% accuracy", doesn't it suffice to simply estimate the average value to be zero?

Here's my reasoning...

  • The function $\sin(x)$, $x \in \mathbb{R}$, is bounded by $-1$ and $1$.
  • The functions $\sin^n(x)$, $x \in \mathbb{R}$, $n \in \mathbb{N}$, are also bounded by $-1$ and $1$. Additionally, as $n$ gets bigger the value of the functions approach zero everywhere except where $\sin(x)$ is exactly equal to $-1$ or $1$, i.e. the limit of this sequence of functions is a function that equals zero "almost everywhere" (I may not be stating that exactly correctly, but I think you get the idea).
  • Over the whole real line, the average of $\sin^n(x)$ approaches zero as $n$ gets bigger. In fact, I think it equals zero for all odd $n$ (and approaches zero from above for even $n$).
  • $n = 100$ is a reasonably big exponent in this context and so it is reasonable to conclude that the average of $\sin^{100}(x)$ is a small positive number (positive because $100$ is even) and estimate it as zero.

So, did I make everything as simple as it can be or did I screw up and make it simpler? (sorry, Einstein)

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The problem statement requires 10% accuracy. Estimating any positive number by zero, however small, is not an estimate within 10%. –  40 votes Jul 24 '13 at 4:57
    
@40 votes I thought you would use $\lvert estimate - actual \rvert$ divided by the size of some range. I further took the range to be the range of the function $\sin^{100}(x)$ or $[0,1]$. The actual values in the other answers are less than $0.08$, so I concluded that an estimate of zero was within $\frac{\lvert 0 - 0.08 \rvert}{\lvert 0 - 1 \rvert} = 0.08$ or $8\%$. –  joeA Jul 24 '13 at 12:08
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@joeA (Old post, sorry.) I would use $|e-a|/|a|$ where $e$ is the estimate and $a$ is the actual value. That is, I think the question is referring to the size of the relative error in the estimate being at most 10%. –  J Swanson Jan 20 at 5:02

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