= Solution
Let $\lambda=\tfrac12\sum_j(x_jdy_j-y_jdx_j)$ on $\mathbb C^3$, so $d\lambda=\omega_{st}$. On a projective line, work in the affine coordinate $w=\rho e^{i\theta}$ and lift it to the unit three-sphere by
$$
s(w)=\frac{(w,1)}{\sqrt{1+|w|^2}}.
$$
This sphere lies in a complex two-plane in $\mathbb C^3$. The defining reduction identity for the <Fubini-Study form> gives $\Omega=s^*\omega_{st}$ on this chart. Differentiating the lift, or using the fact that real radial rescaling contributes no imaginary part to $\bar z\,dz$, gives
$$
s^*\lambda=\frac12\frac{\rho^2}{1+\rho^2}\,d\theta,
\qquad
\Omega=\frac{\rho}{(1+\rho^2)^2}\,d\rho\wedge d\theta.
$$
The missing point at infinity has zero area, so
$$
\boxed{\int_H\Omega=2\pi\int_0^\infty\frac{\rho\,d\rho}{(1+\rho^2)^2}=\pi.}
$$
This normalization is important: it differs from conventions in which a projective line has area $1$ or $2\pi$.
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