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Is the integral $$\int_{0}^{1}dy\int_{1}^{y}e^{-x^2}+e^{x}\sin xdx$$ wrong? I mean in the integral, $0\leq y \leq 1, y\leq x \leq 1 $. Will it lead to a contradiction?

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The convention is that $\int_a^b = -\int_b^a$, so you could replace the $\int_1^y$ by $-\int_y^1$. (But there is no contradiction.) – copper.hat Dec 25 '12 at 7:22
I think so, but rarely do I see this style. – user39843 Dec 25 '12 at 7:24
up vote 2 down vote accepted

There is nothing in principle wrong with it. Recall that $\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx$. Parentheses might be nice for clarity, as in $$\int_{0}^{1}dy\int_{1}^{y}\left(e^{-x^2}+e^{x}\sin x\right)\,dx.$$ If you evaluate it, you will get precisely the negative of $$\int_{0}^{1}dy\int_{y}^{1}\left(e^{-x^2}+e^{x}\sin x\right)\,dx.$$

The shape of the integral is unusual, and it is quite likely that it is not the natural answer to a question.

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I think so. Thx! – user39843 Dec 25 '12 at 7:57

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