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I need to evaluate this integral over the portion of the positive octant cut off by the plane $x + y + z = \pi$. $$\iiint \sin(x+y+z)\, dx\, dy\,dz$$ What boundaries do I use?

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As you are working in the positive octant $$ x \geq 0, y \geq 0, z \geq 0$$

You're other condition is $$ x + y + z \leq \pi $$

Rewriting it: $ z \leq \pi - x - y $

Results in $ z \leq \pi $

Similarly, for any $z$: $ y \leq \pi - x - z \leq \pi - z $

And for any $z$ and $y$: $ x \leq \pi - y - z $

Therefore your boundary conditions are: $$ 0 \leq z \leq \pi $$ $$ 0 \leq y \leq \pi - z $$ $$ 0 \leq x \leq \pi - y - z $$

You're integral then becomes: $$ \int_0^\pi \int_0^{\pi-z} \int_0^{\pi -y-z} \sin (x+y+z)dxdydz$$

Which is equal to $$ \frac{\pi ^2}2 - 2 $$

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The boundaries are

  • $z=0$ and $z(x,y)=\pi-x-y$
  • $y=0$ and $y(x)=\pi-x$
  • $x=0$ and $x=\pi$
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