The problem goes as follows: $$\iiint_{\mathrm{E}}(x^2-z^2)\, dx\,dy\,dz, $$where $\mathrm{E}$ is defined by $x, y,z \ge 0$ and $x+y+z \le 1$. I'm have difficulties finding the limits in order to solve the triple integral. Do I use spherical or cylindrical coordinates? How do I know which of the two to use for future problems? Or does it not matter?
Edit: I have not been able to calculate the integral, any help would be appreciated.
On
Hint. The condition $x, y, z\ge 0$ restricts you to the first octant. Then the equation $x+y+z=1$ defines the plane which intersects the axes at the points $(0, 0, 1), (0, 1, 0)$ and $(1, 0, 0).$ Finally, the inequality $x+y+z\le 1$ says you are interested in the finite region bounded by the first octant and the plane. Can you find your limits now?
Spherical or cylindrical ? Obviously not !
$$E=\{(x,y,z)\in \mathbb R^3\mid x\in [0,1], y\in [0,1-x], z\in [0,1-x-y]\}.$$