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I devised a neat proof that the integral of an odd function over a symmetric interval is $0$. I was immediately tempted to post it to math stack, but feared it would frowned upon as it is not a question. Therefore, I was hoping to find out if there was a place to post non-question based things like a proof.

On another note, if anyone is interested, her is my proof:

Note first that: $$\int_a^b f(x) \, dx = \int_a^b f(a+b-x) \, dx$$ In the case of a symmetric interval, this reads: $$\int_{-a}^a f(x)\,dx = \int_{-a}^a f(a-a-x)\,dx = \int_{-a}^a f(-x) \, dx$$ In the case of an odd function, $f(-x) = -f(x)$, so that $$\int_{-a}^a f(x) \, dx = -\int_{-a}^a f(x) \, dx$$ The only way that a positive number can equal a negative number is if the number is $0$. QED

Michael Hardy
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infinitylord
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  • How else is this proved? – Unit Nov 23 '16 at 01:09
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    That's a pretty standard proof. However, you could post a question and then post the answer to it. $\qquad$ – Michael Hardy Nov 23 '16 at 01:14
  • The first time it was introduced to me, it was displayed graphically/conceptually, so there was no formal proof provided. The next time, it was displayed using u-substitution where $x = -u$. I haven't see this method before, though I understand it's really not much a stretch from the u-sub method, and is unlikely to be original. My question is more-so regarding the policy on math stack for a place to post proofs. – infinitylord Nov 23 '16 at 01:14
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    A blog seems like a good place for this sort of thing. Or the old fashioned version, a journal. – mathematician Nov 23 '16 at 01:15
  • Thank you, Michael, that helps greatly! – infinitylord Nov 23 '16 at 01:15
  • If you aim to ask whether your proof is correct or not, we have the [tag:proof-verification] tag for questions. – shardulc Nov 23 '16 at 01:32
  • ProofWiki is one place to post proofs. – angryavian Nov 23 '16 at 01:36

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