= Solution
Write $H=\int_{\mathbb R}h(x)\,dx$, and assume $0<H<\infty$; a <uniform distribution> on the region requires positive area. Set $x=V/U$, $s=U$, so the inverse transformation is $U=s$, $V=sx$. Its <Jacobian determinant> has absolute value $s$ for $s>0$. The region becomes $x\in\mathbb R$, $0<s\leq\sqrt{h(x)}$, and its area is
$$
|C_h|=\int_{\mathbb R}\int_0^{\sqrt{h(x)}}s\,ds\,dx=\frac H2.
$$
The original <joint probability density function> is therefore $2/H$ on $C_h$. The <change of variables> formula gives
$$
\boxed{f_{X_1,X_2}(x,s)=\frac{2s}{H}\,\mathbf1_{\{0<s\leq\sqrt{h(x)}\}}.}
$$
The boundary $U=0$ has <probability> zero, so division by $U$ causes no ambiguity for the resulting <random variable>.
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