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Integral $\iiint\limits_D \frac{1}{(x+y+z)^3} dxdydz$ should be evaluated. D is area bounded by coordinate planes and $x+y+z=1$ plane.

I need help with determining integration limits. What software would you recommend for drawing 3D objects to develop reasoning for this kind of problems?

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up vote 2 down vote accepted

Write those inequalities out: $$ x \ge 0, y \ge 0 , z \ge 0, x+y+z \le 1 $$ It is clear that the region $D$ is contained within unit box $0 \le x \le 1, 0 \le y \le 1, 0 \le z \le 1$, because if any of $x$,$y$ or $z$ exceeds one, $x+y+z \le 1$ will be violated even if the remaining coordinates are zero.

Now, let's do it step-by-step. Assuming $x$ and $y$ are fixed. Variable $z$ can range, within confines of $D$ from $0$ to $1-x-y$, i.e. $0 \le z \le 1-x-y$.

Assuming $x$ is fixed, variable $y$ satisfies $0 \le y \le 1-x$.

And we have already established that $ 0 \le x \le 1$.

Here is a visualization using Mathematica:

enter image description here

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