Is integration from $a$ to $b$ same or $b$ to $a$ or is negative?

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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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2 Answers

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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.

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Try a substitution $x \mapsto a + b - x$

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