Calculus can be used to sum up a number of small things, so it can speed up adding any sequence.
It can find the area under curves exactly: for instance, it can be used to give you that famous equation for the volume of a sphere.
It gives you a really fast way to get the area of a polygon.
It can be used to calculate the change in something; for instance, if you just measure how fast you're going at regular times, you can calculate the distance you went.
It can also calculate instantaneous speeds. It could be used for radar guns to detect speeding cars.
In my experience it mostly reduces the number of calculations for existing math or gives you an exact answer when you had to approximate before.
