1.13. Find the mass of a square plate, the density of which at each point $M$ is proportional to the sum of the distances from this point to the diagonals of the plate.
The problem is solved. The plate is depicted in Fig. 1.37 in the chosen Cartesian coordinate system.

Given $p(x, y)=K(x+y)$, therefore

$$
\begin{gathered}
M(\Omega)=\iint_{\Omega} K(x+y) d x d y=4 K \int_{0}^{a} d x \int_{0}^{a-x}(x+y) d y= \\
=\left.4 K \int_{0}^{a}\left(x y+\frac{y^{2}}{2}\right)\right|_{0} ^{a-x} d x=4 K \int_{0}^{a}\left(x(a-x)+\frac{1}{2}(a-x)^{2}\right) d x= \\
=\left.4 K\left(a \frac{x^{2}}{2}-\frac{x^{3}}{3}-\frac{1}{6}(a-x)^{3}\right)\right|_{0} ^{a}=\frac{4}{3} K a^{3}
\end{gathered}
$$