How can I calculate $\int^{b}_{a}\Phi\left(\frac{x-\mu}{\sigma}\right)\phi(x)\,\mathrm dx$ Suppose $\phi(\cdot)$ and $\Phi(\cdot)$ are the density function and cumulative distribution function of the standard normal distribution. $a<b$ are finite real numbers.
How can I calculate the integral:
$$\int^{b}_{a}\Phi\left(\frac{x-\mu}{\sigma}\right)\phi(x)\,\mathrm dx$$
Note that the question is similar to this one. There, the comments to the accepted answer suggest that the solution can be generalized to definite integrals, but I haven't been able to figure it out.
 A: Partial answer.  If I have some time, I will work on tightening up the result.
In the other (non accepted) answer to the question cited by the OP, it is shown that if $X\sim N(\mu,\sigma^2)$ and $Y\sim N(0,1)$ are independent normal random variables, then 
\begin{align}
P\{X \leq Y \mid Y = w\} &= P\{X \leq w \mid Y = w\}\\
&= \int_{-\infty}^w f_{X\mid Y}(t\mid Y=w) \,\mathrm dt\\
&= \int_{-\infty}^w f_{X}(t) \,\mathrm dt &\scriptstyle{\text{since $X$ and $Y$ are independent}}\\
&= \Phi\left(\frac{w-\mu}{\sigma}\right).
\end{align}
Consequently, 
\begin{align}
\int_a^b \Phi\left(\frac{w-\mu}{\sigma}\right)\phi(w) \,\mathrm dw
&= \int_a^b \left[\int_{-\infty}^w f_{X}(t) \,\mathrm dt\right]f_Y(w)\,\mathrm dw\\
&= \int_{w=a}^{w=b}\int_{-\infty}^w f_{X,Y}(t,w)\,\mathrm dt \,\mathrm dw.
\end{align}
The value of this double integral is the probability that $(X,Y)$ lies in a horizontal strip of width $b-a$ extending from $-\infty$ to the points on the line segment with end-points $(a,a)$ and $(b,b)$ in the $t$-$w$ plane. The computation of the exact value is not easy but at least we can bound the value fairly easily. 
Since $\Phi\left(\frac{w-\mu}{\sigma}\right)$ increases from $\Phi\left(\frac{a-\mu}{\sigma}\right)$ to $\Phi\left(\frac{b-\mu}{\sigma}\right)$ as $w$ increases from $a$ to $b$, we get the bounds
$$\Phi\left.\left.\left(\frac{a-\mu}{\sigma}\right)\right(\Phi(b)-\Phi(a)\right)
< \int_a^b \Phi\left(\frac{w-\mu}{\sigma}\right)\phi(w) \,\mathrm dw < \Phi\left.\left.\left(\frac{b-\mu}{\sigma}\right)\right(\Phi(b)-\Phi(a)\right)$$
which, depending on the values of $a,b,\mu$ and $\sigma$ might be tight enough to be satisfactory for gummint purposes if not for stats.SE purposes.
