 Solution

Let $h\ge 0$ be proportional to the target probability density function, with $0<Z=\int h(x)dx<\infty$. Define the ratio-of-uniforms method through the regionDraw $(U,V)$ uniformly with respect to area on $A_h$ and return $X=V/U$. To prove its law, change variables from $(u,x)$ to $(u,v)=(u,ux)$. The Jacobian determinant is $u$, and the allowed interval for $u$ is $0<u\le \sqrt{h(x)}$. ConsequentlyThe joint density of $(U,X)$ is therefore $2u/Z$ on that interval. Integrating out $u$ givesFor an already normalized density, $Z=1$ and the region has area $1/2$. The name refers to the two coordinates of a uniformly sampled region; drawing two unrelated uniforms on arbitrary intervals and taking their ratio would not give the target law.