Expected minimum of two independant random variables

Question

This is part of an alternative problem of the newsvendor model.
Let $D_1$ and $D_2$ be two independant and positives random variables and let $f_{D_1}$ and $f_{D_2}$ be their density functions.
I need to minimize the function $q\longmapsto\mathbb{E}\left[\mathrm{min}(D_2,(q-D_1)^+)\right]$
I started by calculating $$\begin{aligned} \mathbb{E}\left[\mathrm{min}(D_2,(q-D_1)^+)\right]&=\int\int_\mathbb{R^2} min(y,(q-x)\mathbb{1}_{x<q})f_{D_1}(x)f_{D_2}(y) \ dx dy\\ &= \int_{x=0}^q\left(\int_{y=0}^{q-x}yf_{D_2}(y)dy\right) f_{D_1}(x)dx + \int_{x=0}^q \left(\int_{y=q-x}^{+\infty}f_{D_2}(y)dy\right)(q-x)f_{D_1}(x)dx \end{aligned}$$ but at that point I think I messed up because I couldn't come up with something nice. I wonder if there is another way to calculate this expression., using cumulative distribution functions or conditional expectations.