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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.)2012-12-25
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    I think so, but rarely do I see this style.2012-12-25

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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!2012-12-25