# Triple integral integration limits

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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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 $0 \le x \le 1$.