The integration is generally area under the curve in $\Bbb R^2 \to \Bbb R$
Is integration in the range from $a$ to $b$ the same as $b$ to $a$ or is it negative?
If it is negative, Is it merely notion of convention or is there some intuition for it?
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2Integration is generally not area under a curve. Areas are positive, whereas integrals can be negative. – joriki Nov 07 '12 at 22:19
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@joriki Ok, indefinite intrgrals are definitely not area under the curve. Do definite integrals represent one? If they are even definite integrals(area) turns out are negative right? – 007resu Nov 08 '12 at 19:09
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I meant definite integrals. Unfortunately I can't understand your last sentence; please try rephrasing it. – joriki Nov 08 '12 at 21:49
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if that's definite, you mentioned, then my last sentence is not important. Even wrong. Thank you. :) – 007resu Nov 08 '12 at 22:55
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Suppose that $f$ and $F$ are functions such that $F'=f$. By Fundamental Theorem of Calculus, we have $$\int_b^af(x)\,dx=F(a)-F(b)=-\bigl(F(b)-F(a)\bigr)=-\int_a^bf(x)\,dx.$$ This approach doesn't work for all integrable functions $f$ (there may be no such $F$), but it at least gives us an idea.
Cameron Buie
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