Define an integral as follows
$$\small{h(\theta)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}[f(r_1)(1-f(r_1\cos\theta+r_2\sin\theta))+f(r_1\cos\theta+r_2\sin\theta)(1-f(r_1))]\phi(r_1)\phi(r_2)dr_1dr_2}$$ where $\phi(r)$ is the probability density function of standard normal distribution, and $f$ is \begin{equation} \begin{aligned} f(x)&=1,\ &&x\geq s\\ &=\frac{1}{2}+\frac{x}{2s},\ &&x\in (-s,s)\\ &=0,\ &&x\leq -s \end{aligned} \end{equation}
(1) I have tried the case of $s=0$, so that \begin{equation} \begin{aligned} f(x)&=1,\ x\geq 0\\ &=0,\ x< 0 \end{aligned} \end{equation} The integral equals to $\frac{\theta}{\pi}$.
(2) However, I was not able to compute for the case of $s\neq 0$. Is it possible to compute this integral analytically in this case?
Many thanks for all comments and helps.