I'm struggling to find limits of integration when the outermost integral depends on $x$ and $z$.
For example, when we study the shadow of the region on the $xz$-plane, we find that it's a square. But its projection upwards (in the direction of the positive $y$ axis) depends on two different surfaces (the cylinder $1-y^2$ and the plane $1-y$), and I don't know how to express everything in terms of one variable.
How can I approach these kinds of exercises?
Edit: my textbook says we should verify the following equalities
$$z = 1 - y \iff y = 1- z \\ x = 1 - y^2 \ \ \underset\iff{y\geq0} \ \ y = \sqrt{1-x}$$
Therefore
$$x = z^2 - 2z \\ z=1-\sqrt{1-x}$$
But what are these surfaces bounds of? I'm really confused.
Simplest (and easiest) way to approach this kind of problems is to look straight at the limits of integration.
The domain can be described by the following inequalities. $$0\leq y \leq 1,\ 0\leq x \leq 1-y^2,\ 0\leq z\leq 1-y $$ Those are equivalent to the following $$0\leq z\leq 1, \ 0\leq x\leq 1,\ y^2 \leq 1-x, \ 0\leq y\leq 1-z $$ Well, the last two inequalities can be written as $0\leq y\leq \min (\sqrt{1-x},\ 1-z)$. This leaves us with $$0\leq z\leq 1,\ 0\leq x\leq 1,\ 0\leq y\leq \min (\sqrt{1-x},\ 1-z) $$ and the integral can be written as $$\int_0^1 \int_0^1 \int_0^{\min (\sqrt{1-x},\ 1-z)} dydxdz. $$