Example 1. Find the mass of the body $\Omega$ with density $\mu=2 x$, bounded by the surfaces

$$
x=2 \sqrt{2 y}, \quad x=\sqrt{2 y}, \quad z=1-y, \quad z=0
$$
Solution.

1. The mass of the body $\Omega$ with density $\mu=2 x$ is determined by the formula

$$
m=\iiint_{\Omega} 2 x d x d y d z
$$

2. Let's define the region $\Omega$ using inequalities. Since $2 \sqrt{2 y} \geq \sqrt{2 y}$, for $x$ we have the inequalities $\sqrt{2 y} \leq x \leq 2 \sqrt{2 y}$. Since $y$ appears under the square root, $y \geq 0$. For $z$, the possible inequalities are $0 \leq z \leq 1-y$ or $1-y \leq z \leq 0$. In the first case, $0 \leq y \leq 1$. In the second case, $y \geq 1$, i.e., the region is unbounded, which is unacceptable.

Thus,

$$
\Omega=\left\{\begin{array}{cc} 
& \sqrt{2 y} \leq x \leq 2 \sqrt{2 y} \\
(x, y, z): & 0 \leq y \leq 1 \\
& 0 \leq z \leq 1-y
\end{array}\right\}
$$

3. We compute $m$ by reducing the triple integral to a repeated integral:

$$
m=\iiint_{\Omega} 2 x d x d y d z=\int_{0}^{1} d y \int_{\sqrt{2 y}}^{2 \sqrt{2 y}} 2 x d x \int_{0}^{1-y} d z=1
$$

Answer. $m=1$ unit of mass.