When you apply the work energy theorem i.e.
$$ \int F(x) dx = \Delta E_{\mathrm{kin}} $$
you typically know the initial speed and the force distance relation $F(x)$ and then wants to calculate the final speed. For example in calculating the speed of an arrow of a compound bow. There are many examples like that.
The "impulse momentum" theorem (i.e. newtons second law in integral form) has a similar mathematical structure but integrating over time instead:
$$ \int F(t) d t = \Delta p $$
What are (real world) examples where you know $F(t)$ (analytically or just graphically) and want to calculate $\Delta p$ (or $\Delta v$)?
The only example I found so far was to calculate the take off speed in jumping in biomechanics there you measure $F(t)$ by using a force platform. However I don't see in this case why not measuring the take of speed directly (using a light barrier, or using video analysis, with a slow motion camera or just calculating it from flight time or height), so the example doesn't really count.