Calculate the static moment with respect to the x-axis

Calculate the static moment with respect to the $x$-axis of the arc $y=\cos x$ between $x=-\dfrac\pi2$ and $x=\dfrac\pi2$.

Step 1: Arc-length element

\[y'=-\sin x \implies ds=\sqrt{1+\sin^2x}\,dx\]

Step 2: Set up and substitute

\[M_x=\int_{-\pi/2}^{\pi/2}\cos x\sqrt{1+\sin^2x}\,dx\]

With $u=\sin x$: \[M_x=\int_{-1}^1\sqrt{1+u^2}\,du = 2\int_0^1\sqrt{1+u^2}\,du\]

Step 3: Evaluate

\[M_x=2\left[\frac{u\sqrt{1+u^2}}2+\frac12\ln\left(u+\sqrt{1+u^2}\right)\right]_0^1 = \sqrt2+\ln(1+\sqrt2)\]

\[\boxed{M_x=\sqrt2+\ln(1+\sqrt2)\approx2.295}\]

Step 4: Note

This is numerically the arc length of the same curve (Problem 106's underlying integral form appears here too), which is expected: for this particular arc, $y\,ds$ and $ds$ happen to reduce to the identical standard integral $\sqrt{1+u^2}\,du$ after the substitution.