Find the static moment (first moment) with respect
Find the static moment (first moment) with respect to the coordinate axes of the line segment $x+y=1$ between the coordinate axes.
Explanation of Method
Parametrize the segment as $y=1-x$, form the constant arc-length element $ds=\sqrt2\,dx$, and integrate $y\,ds$ and $x\,ds$ over $x\in[0,1]$.
Step 1: Endpoints and arc-length element
The segment runs from $(1,0)$ to $(0,1)$; with $y=1-x$, $ds=\sqrt{1+1}\,dx=\sqrt2\,dx$.
Step 2: Moment about the $x$-axis
\[M_x=\int_0^1(1-x)\sqrt2\,dx = \sqrt2\left[x-\frac{x^2}2\right]_0^1=\sqrt2\cdot\frac12=\frac{\sqrt2}2\]
Step 3: Moment about the $y$-axis
\[M_y=\int_0^1x\sqrt2\,dx=\sqrt2\left[\frac{x^2}2\right]_0^1=\frac{\sqrt2}2\]
Step 4: Centroid check
Total length $L=\sqrt2$, so $\bar x=\bar y=\dfrac{\sqrt2/2}{\sqrt2}=\dfrac12$ — the segment's midpoint, as expected by symmetry.
\[\boxed{M_x=M_y=\frac{\sqrt2}2}\]
Result
Final answer:
\[\boxed{M_x = M_y = \frac{\sqrt{2}}{2}}\]