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One of the major difficulties of student in advanced calculus (including myself when student) is to obtain the extremes of repeated integrals to calculate the volume integral in $R^n$ i.e. transform the integral $$ \int_{D}f(x) \,\mathrm d\,V_n, \quad f\in C(\mathbb{R}^n) $$ in the multiple integral $$ \int_{c_1^1}^{d_1^1}\cdots\int_{c_n^1}^{d_n^1} f(x_1,\ldots,x_n) \,\mathrm d\,x_1\ldots \,\mathrm d\,x_n +\ldots + \int_{c_1^m}^{d_1^m}\cdots\int_{c_n^m}^{d_n^m} f(x_1,\ldots,x_n) \,\mathrm d\,x_1\ldots \,\mathrm d\,x_n. $$ The integer $m$ arises from the need to partition the set $D$.

More precisely I would get for a well-defined algorithm (that could be manually computable) whose output was the number $m\in\mathbb{N}$ and the extremes of integration in terms of vectors $a\in\mathbb{R}^p$, $b\in\mathbb{R}^p$ ​​and the matrices $M^1,\ldots,M^p\in\mathbb{R}^{n\times n}$ that define the set $D\subset\mathbb{R}^n$ as below $$ D= \left\{ x= \left[ \begin{array}{c} x_1 \\ \vdots \\ x_n \end{array} \right] \in\mathbb{R}^n\quad \left|\quad \begin{array}{c} a_1 \leq x^TM^1x \leq b_1 \\ \vdots\\ a_p \leq x^TM^px \leq b_p \\ \end{array} \right. \right\} \subset [-r,r]^n,\quad r>0. $$

In terms of explicit coordinates we have: $M^1=(M_{uv}^1)_{n\times n},\ldots ,M^p=(M_{uv}^p)_{n\times n}$ are the matrices in $\mathbb{R}^{n\times n}$ and $a=(a_i)_p$ and $b=(b_j)_p$ are the vectors in $\mathbb{R}^{p}$.

My question: Is there any algorithm that can be implemented but it is computationally simple for calculate the extrems of repeated integrals above? Some reference?

Thanks in advance.

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computationally simple for computer or for human? – Norbert May 6 '13 at 20:10
@Norbert For human. – MathOverview May 7 '13 at 14:29

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