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I noticed that the horizontal pivot line (or $y$-coordinate of the centroid) under the curve $y=\sin^2 x$ between $0$ and $\pi$ is exactly $\frac{3}{8}$. There may be no reason for me to find this strange, but it's just so neat. Does anyone know why this is?

$$\frac{1}{2}\frac{\int_0^{\pi} (\sin^4 x) dx}{\int_0^{\pi} (\sin^2x) dx} = \frac{3}{8}.$$

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    Pardon me, but What is the definition of "horizontal pivot line"?2012-02-11
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    The OP is presumably referring to the $y$-coordinate of the centroid. "Why" is a question worth thinking about. The usual argument for the location of the centroid tells us **that** the $y$-coordinate is given by the integral. There may be a more intuitive argument.2012-02-11
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    When finding the centre of mass of a given area under a curve, you would do so by finding the horizontal and vertical 'pivot lines'.2012-02-11
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    @AndréNicolas exactly. :)2012-02-11
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    Thanks for the clarification.2012-02-11
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    Good instincts: it *is* interesting that the $y$-coordinate of the centroid is such a ‘nice’ numbers.2012-02-11
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    @Brian, Korgan: Can you explain what it is that you find so neat about the number $\frac38$? Is it just that it is rational and doesn't involve $\pi$?2012-02-11
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    @Rahul: It is at first blush a little surprising to get such a simple rational out of a context in which $\pi$ plays a prominent rôle. It’s perhaps rather less surprising after one has some experience, but I think that a beginning student should be commended for wondering whether there’s a simple, intuitive explanation.2012-02-11

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