Evaluate the triple integral $\iiint (x^2+y^2+z^2)\,dx\,dy\,dz$ I have to evaluate this integral. It is enough for me to know the correct limits to integration.
$$ \iiint_W  (x^2 + y^2 + z^2) \,\mathrm dx\,\mathrm dy\,\mathrm dz$$
Conditions: 
$$x\ge 0,\quad y \ge 0 ,\quad z \ge 0,\quad 0 \le x + y + z \le a,\quad (a>0)$$
My ideas:
As I have 
$0 \le x + y + z \le a$,
I can say that $$0 \le x + y + z$$
$$x \le - z - y $$
so my limits to integration could be: 
$$\int _{ z }^{ a }{  } \int _{ -z-y }^{ y }{  } \int _{ 0 }^{ -z-y }{ (x^2 + y^ 2+ z^ 2) \,\mathrm dx\,\mathrm dy\,\mathrm dz } $$
Can somebody help/correct me?

Your integral should be of the form
$$\int_{z_0}^{z_1}\int_{y_0(z)}^{y_1(z)}\int_{x_0(y,z)}^{x_1(y,z)}(x^2+y^2+z^2)\,\mathrm dx\,\mathrm dy\,\mathrm dz $$
By the conditions, we find $0\le z\le a$, and for given $z$, $0\le y\le a-z$, and for given $z$ and $y$ $0\le x\le a-y-z$. So in the end it should be 
$$\int_{0}^{a}\int_{0}^{a-z}\int_{0}^{a-z-y}(x^2+y^2+z^2)\,\mathrm dx\,\mathrm dy\,\mathrm dz $$
