In a game, one of the most basic systems you may come across is Euler's Method. It's a good way of approximating the result of an integration over time. Here is a simple example:
Velocity = Velocity + (Acceleration * ElapsedTimeSinceLastUpdate);
Position = Position + (Velocity * ElapsedTimeSinceLastUpdate);
Bare in mind that integrating a function of acceleration gives you the function of velocity and integrating the function of velocity gives you the function of position.
This is useful for doing many actions over a period of time.
One of the more common ways I have used calculus however, is simply to prove that an equation I have come up with will work as I intend before I go to the trouble of implementing it.
For example, I once made an equation of velocity which would change an objects orientation over a fixed time period until it had moved to a new desired orientation (it was a bell curve so it would ease in and out of the animation).
By integrating the equation over the fixed period I could show that it would always result in the total change in angle I required. This meant that, with some simple calculus on paper, I could be sure it was worth implementing before I went ahead. It also meant that I had an equation of position which could simply be queried relative to the time since the start of the animation so I wouldn't have to constantly add the velocity onto the position.