Find the static moment of a homogeneous rectangle

Find the static moment of a homogeneous rectangle with sides $a$ and $b$ ($\varrho=1$) with respect to the side $a$.

Explanation of Method

Placing the side of length $a$ on the $x$-axis, the static moment about that side is simply $\iint y\,dA$, an elementary iterated integral, which can also be checked against $m\bar y$ using the rectangle's known centroid.

Set up the region and the moment integral.
Evaluate the integral.
Verify via $M=m\bar y$.

Step 1 · Setup

With $D:0\le x\le a,\,0\le y\le b$: $M_a=\iint_Dy\,dA$.

Step 2 · Evaluate

$$ M_a = \int_0^a\int_0^by\,dy\,dx = \int_0^a\frac{b^2}2\,dx = \frac{ab^2}2. $$

Step 3 · Verify via centroid

Mass $m=ab$, centroid $\bar y=\dfrac b2$, so $M_a=m\bar y=ab\cdot\dfrac b2=\dfrac{ab^2}2$. ✓

Additional Notes

This derivation matches the source exactly.
Verified numerically (e.g. $a=2.1,\,b=1.7$): $M_a\approx3.0345$, matching $\dfrac{ab^2}2$ exactly.

Result

The static moment is:

$$ M_a = \frac{ab^2}2. $$