Find the static moment with respect to the x-axis

Find the static moment with respect to the $x$-axis of the area bounded by $y=\dfrac{2}{1+x^2}$ and $y=x^2$.

Step 1: Find the intersections

Setting $\dfrac2{1+x^2}=x^2$ gives $x^4+x^2-2=0$, i.e. $(x^2+2)(x^2-1)=0$, so $x^2=1$ (the only valid root) and $x=\pm1$, meeting at height $y=1$.

Step 2: Identify the upper curve

At $x=0$: $y=2$ for the rational curve vs.\ $y=0$ for the parabola, so $\dfrac2{1+x^2}\ge x^2$ throughout $[-1,1]$.

Step 3: Set up the moment integral

\[M_x=\int_{-1}^1\int_{x^2}^{2/(1+x^2)}y\,dy\,dx = \frac12\int_{-1}^1\left[\frac4{(1+x^2)^2}-x^4\right]dx\]

Step 4: Evaluate the rational term

Using $\displaystyle\int\frac1{(1+x^2)^2}\,dx=\frac x{2(1+x^2)}+\frac12\arctan x+C$:

\[\int_{-1}^1\frac4{(1+x^2)^2}\,dx = 2+\pi\]

Step 5: Evaluate the polynomial term

\[\int_{-1}^1x^4\,dx=\frac25\]

Step 6: Combine

\[M_x=\frac12(2+\pi)-\frac12\cdot\frac25=\frac{2+\pi}2-\frac15=\frac{8+5\pi}{10}\]

\[\boxed{M_x=\frac{8+5\pi}{10}\approx2.371}\]