858. Find the mass of the body bounded by the cylindrical surface $x^{2}=2 y$ and the planes $y+z=1, 2 y+z=2$, if at each point of the body the volumetric density is numerically equal to the ordinate of that point.
Solution. According to the condition, at point $M(x, y, z)$ of the body, the volume density $\delta(M)=y$. By formula (2), the mass of this body is

$$
m=\iint_{G} \delta \delta(M) d v=\iiint_{G} y d x d y d z
$$

where $G$ is the region occupied by the given body (Fig. 182).[^31]

Evaluating the triple integral using formula (*), we obtain:

$$
\begin{aligned}
& m=\iint_{i_{x y}} y d x d y \int_{1-y}^{2(1-y)} d z=\iint_{A(/ / i} y(1--y) d x d y=\int_{n}^{1}\left(y-y^{2}\right) d y \int_{-V^{-\overline{2 y}}}^{\sqrt{-1-1}} d x= \\
& =\int_{0}^{1}\left(y-y^{2}\right) 2 \sqrt{2 y} d y=\left.2 \sqrt{2}\left(\frac{2}{5} y^{\frac{5}{2}}-\frac{2}{7} y^{\frac{7}{2}}\right)\right|_{0} ^{1}=\frac{8 \sqrt{\bar{z}}}{35} .
\end{aligned}
$$