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\begin{align}
1 &= K \int_{-\pi/2}^{\pi/2}\cos^{2}\pars{\theta}
{\rm g}\pars{KR\sin\pars{\theta}}\,\dd\theta\
\\[3mm]&\stackrel{R \sim 0}{\sim}\ K \int_{-\pi/2}^{\pi/2}\cos^{2}\pars{\theta}
\braces{{\rm g}\pars{0} + {\rm g}'\pars{0}KR\sin\pars{\theta}
+
\half\,{\rm g}''\pars{0}\bracks{KR\sin\pars{\theta}}^{2}}\,\dd\theta
\\[3mm]&=K{\rm g}\pars{0}
\underbrace{\int_{-\pi/2}^{\pi/2}\cos^{2}\pars{\theta}\,\dd\theta}
_{\ds{=\ {\pi \over 2}}}
+
{1 \over 8}\,K^{3}{\rm g}''\pars{0}R^{2}
\underbrace{\int_{-\pi/2}^{\pi/2}\sin^{2}\pars{2\theta}\,\dd\theta}_{\ds{=\ {\pi \over 2}}}
\end{align}
$\color{#00f}{\large\ds{K = {2 \over \pi{\rm g}\pars{0}}}}$