Determinate the limits of integration of $$\iiint_D f(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z,$$
where $D$ is the region delimited by $x=0, y=0, z=x+y, z=1$.
My try
I'm having trouble determinating the limits of integration in the $y$ variable. I determinated that the limits on $x$ are $x_1=0$ and $x_2=z-y$ and the limits in $z$ are $z_0=0$ and $z_1=1$, but I don't know how to determinate the limits for $y$.
Any hints are appreciated.
The way to approach this type of problem is to see how the domain of integration looks like. $x=0$ and $y=0$ are two perpendicular planes containing the origin. The third perpendicular plane through origin wold be $z=0$. But you have a plane at $z=1$ instead. $z=x+y$ is a tilted plane through the origin, since $(0,0,0)$ is on the plane. That means your domain $D$ is a tetrahedron. If you look at the intersection of the tilted plane with $z=1$ you get $x+y=1$. When $x=0$ you get $y=1$, and when $y=0$ you get $x=1$. So the vertices of the tetrahedron are $(0,0,0)$, $(0,0,1)$, $(1,0,1)$, and $(0,1,1)$.
Now the limits of integration depend on the integration order. So I can write for example $$\int_0^1 dx\int_0^{1-x}dy\int_{x+y}^1 dz\ f(x,y,z)$$