Limits of Integration Over Iterated Region For $\epsilon>0$ and $y\gg\epsilon$ I have an integral on the region $$P(n,y)=\left\{(x_1,\ldots,x_n)\in\mathbb{R}^n\;:\; \sum_k^nx_k\geq y, \;x_i\geq \epsilon, \forall i \right\}.$$
    I want to explicitly write down the limits of integration when the integral is broken down. For example, for $n=1,2$
    $$\int_{P(1,y)}f(x)d(x)=\int_y^{\infty}f(x)dx$$
    $$\int_{P(2,y)}f(x)d(x) = \int_{\epsilon}^{y-\epsilon}\int_{y-x_2}^{\infty}f(x_1,x_2)dx_1dx_2+ \int_{y-\epsilon}^{\infty}\int_{\epsilon}^{\infty}f(x_1,x_2)dx_1dx_2$$
Up to $n=3$ it is easy to visualize. The region $P(3,y)$ is just getting everything except a small ``triangle" near the origin. In the lower level, each slice is simply the region $P(2,\cdot)$, but after the plane $x_1+x_2+x_3=\epsilon$ intersects the $x_3$-axis you just get a giant cube. With this, I have written the regions for $P(3,y)$ as 
        $$\int_{P_3}dx_1dx_2dx_3=\int_{\epsilon}^{y-2\epsilon}\int_{P(2,\epsilon-u_3)}dx_1dx_2dx_3+\int_{y-2\epsilon}\int_{\epsilon}^{\infty}\int_{\epsilon}^{\infty}dx_1dx_2dx_3.$$
    My goal is to write out the integral without any $P_n$'s. For $n=3$, the reasoning above leads to the explicit limits
    $$\int_{P_3}=\int_{\epsilon}^{y-2\epsilon}\int_{\epsilon}^{y-x_3-\epsilon}\int_{y-x_1-x_2}^\infty+\int_{\epsilon}^{y-2\epsilon}\int^{\infty}_{y-x_3-\epsilon}\int_{\epsilon}^\infty+\int^{\infty}_{y-2\epsilon}\int_{\epsilon}^{\infty}\int_{\epsilon}^\infty$$
For general $n\geq 2$, I am looking for something like
        $$\int_{P_n}dx_1\ldots dx_n=\int_{y-(n-1\epsilon)}^\infty\underbrace{\int_\epsilon^\infty \cdots\int_\epsilon^\infty}_{n\text{ times}}dx_1\ldots dx_n + \int_{\epsilon}^{y-(n-1)\epsilon}``(n-1)\text{ integrals"}dx_1\ldots dx_n$$
Any help finding the $n-1$ ``integrals?" Thanks, 
UPDATE: I think I found something much simpler. You can write $$\int_{x_1,\ldots,x_n\geq\epsilon}dx_1\ldots dx_n=\int_{P(n,y)}dx_1\ldots dx_n+\int_{D(n,y)}dx_1\ldots dx_n,$$ with $D(n,y)=\left\{(x_1,\ldots,x_n)\in\mathbb{R}^n\;:\; \sum_k^nx_k< y, \;x_i\geq \epsilon, \forall i \right\}$. Note the region $D$ is the ``triangle" part I was talking about. The limits over that region for general $n$ are 
$$\int_{D(n,y)}dx_1\ldots dx_n=\int_{\epsilon}^{y-(n-1)\epsilon}\int_{\epsilon}^{y-(n-2)\epsilon-x_{n-1}}\int_{\epsilon}^{y-(n-3)\epsilon-x_{n-1}-x_{n-2}}\cdots\int_{\epsilon}^{y-\sum_{i=1}^nx_i}dx_1\ldots dx_n$$ Use these two results to rewrite the integral over $P(n,y)$.
 A: $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
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 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
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$\ds{y \gg \epsilon > 0}$.

\begin{align}
&\int_{\ds{\mathbb{R}^{n}}}\mrm{f}\pars{x_{1},\ldots,x_{n}}
\bracks{\sum_{k = 1}^{n}x_{k} > y}\prod_{j = 1}^{n}\bracks{x_{j} > \epsilon}\,
\dd x_{1}\ldots\, x_{n}
\\ = &\
\int_{\ds{\mathbb{R}^{n}}}\mrm{f}\pars{x_{1},\ldots,x_{n}}
\prod_{j = 1}^{n}\bracks{x_{j} > \epsilon}\
\overbrace{\int_{0^{+} - \infty\ic}^{0^{+} + \infty\ic}{\exp\pars{ \braces{\sum_{k = 1}^{n}x_{k} - y}s} \over s}\,{\dd s \over 2\pi\ic}}
^{\ds{=\ \bracks{\sum_{k = 1}^{n}x_{k} > y}}}\ \,\dd x_{1}\ldots\, x_{n}
\\[5mm] = &\
\int_{0^{+} - \infty\ic}^{0^{+} + \infty}{\expo{-ys} \over s}
\bracks{\int_{\epsilon}^{\infty}\cdots\int_{\epsilon}^{\infty}
\mrm{f}\pars{x_{1},\ldots,x_{n}}\exp\pars{\bracks{x_{1} + \cdots + x_{n}}s}
\,\dd x_{1}\ldots\dd x_{n}}\,{\dd s \over 2\pi\ic}
\end{align}

$\ds{\large Indeed}$, this expression is useful whenever we know an explicit form of$\ds{\,\mrm{f}\pars{x_{1},\ldots,x_{n}}}$. Namely, it avoids 'to worry' for the integration limits.

