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Let $f \colon \mathbb{R}^n \to \mathbb{R} $ and $A=\{x \in [0,1]^n\colon x_1 \leq x_2 \leq \cdots \leq x_n\}$ I wan to compute a integral of the form $$I =\int_A \prod_{i=1}^n f(x_i -x_{i-1})dx = \int_{0}^1\int_{0}^{x_n} \cdots \int_0^{x_2}\prod_{i=1}^n f(x_i -x_{i-1})dx_1 \cdots dx_n$$ With the change of variables $y = F(x)$, $F_i(x)=x_i-x_{i-1}$ ($x_0=0$) we have $$I=\int_{F^{-1}(A)}\prod_{i=1}^n f(y_i)dy, \qquad F_i^{-1}(y) = y_1 + \cdots + y_i$$ Now I'm trying to know the set $F^{-1}(A)$ so I can have a explicit form for the limits of integration.

Any help will be appreciated.

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$$ F^{-1}(A)=\{y\in\mathbb{R}^n\colon y_j\ge0,\; j=1,\dots, n,\; \sum_j y_j\le 1\}. $$

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