Find the static moment with respect to the coordinate

Find the static moment with respect to the coordinate axes of the triangle bounded by $x+y=1$, $x=0$, and $y=0$.

Step 1: Region

Right triangle with vertices $(0,0),(1,0),(0,1)$: $0\le x\le1,\ 0\le y\le1-x$.

Step 2: Moment about the $x$-axis

\[M_x=\int_0^1\int_0^{1-x}y\,dy\,dx = \int_0^1\frac{(1-x)^2}2\,dx = \frac12\left[x-x^2+\frac{x^3}3\right]_0^1=\frac16\]

Step 3: Moment about the $y$-axis

\[M_y=\int_0^1\int_0^{1-x}x\,dy\,dx = \int_0^1x(1-x)\,dx = \left[\frac{x^2}2-\frac{x^3}3\right]_0^1=\frac16\]

Step 4: Centroid check

Area $A=\frac12$, so $\bar x=\bar y=\dfrac{1/6}{1/2}=\dfrac13$ — the standard result that a triangle's centroid lies one-third of the way from a side along each median from the opposite vertex.

\[\boxed{M_x=M_y=\frac16}\]