Example 6. Find the static moments with respect to the coordinate planes and the coordinates of the center of gravity of a homogeneous triangular plate $x+y+z=a, x \geqslant 0, y \geqslant 0, z \geqslant 0$.
Solution. The mass of the plate can be easily found geometrically. It coincides with the area of an equilateral triangle with side $\sqrt{2} a: m=s=\frac{a^{2} \sqrt{3}}{2}$. Since this triangle is equilateral, the static moments relative to the coordinate planes are equal:

$$
\begin{aligned}
M_{x y}=M_{y z}= & M_{x z}=\iint_{\sigma} z d \sigma=\int_{\substack{0 \leqslant x \leqslant a \\
0 \leqslant y \leqslant a-x}}(a-x-y) \cdot \sqrt{3} d x d y= \\
& =\sqrt{3} \int_{0}^{a} d x \int_{0}^{a-x}(a-x-y) d y=\frac{1}{2} \sqrt{3} \int_{0}^{a}(a-x)^{2} d x=\frac{\sqrt{3}}{6} a^{3}
\end{aligned}
$$

Therefore, $x_{c}=y_{c}=z_{c}=\frac{a}{3}$.

$$
\text { Answer. } \quad M_{x y}=M_{y z}=M_{x z}=\frac{\sqrt{3}}{6} a^{3}, x_{c}=y_{c}=z_{c}=\frac{a}{3} \text {. }
$$