Calculate the integral $$ \iiint_K (x^2 -z^2)\, dx\, dy \,dz\,, $$ where $$K=\left\{(x,y,z):x,y,z\ge 0,x+y+z\le 1\right\}$$
I have tried solving it but I really don't understand how to handle the upper limits. I really need some guidance.
Update: I got it to zero by using it's symmetry as given in comments. Thanks for the help. I also confirmed it by calculating it myself with the values inputted.
The region has an $xz$ exchange symmetry, i.e.
$$(x,y,z) \in K \implies (z,y,x) \in K$$
In other words
$$I = \iiint_K f(x,y,z)\:dV = \iiint_K f(z,y,x)\:dV$$
but in this problem for $f(x,y,z) = x^2-z^2$ we have that
$$f(z,y,x) = -f(x,y,z)$$
which means
$$I = -I \implies I = 0$$