Check out this article in the New York Times!
Dr. Mathieu Ossendrijver from Humbolt University translated cuneiform tablets dating around 350-50 B.C., describing how Babylonian astronomers calculated the distance that the planet Jupiter traveled across the sky. They did this by recording its velocity at specific times, and then approximating the area under this graph — i.e. approximating the definite integral.
If you haven’t taken calculus yet, the process is still understandable. We know that distance traveled = speed x time; for example, if you travel at 60 miles/hr for 2 hours, you have traveled 120 miles. This works if you keep your speed roughly the same. On the other hand, if your speed keeps changing, you could instead calculate this on small time intervals, and then add them all up to get the total distance traveled. That is what the Babylonians did.
We’ve known many ancient cultures — China, Greece, etc. — used versions of what we now call the definite integral to calculate areas of shapes. However, this is the earliest documented instance of where the “shape” was a velocity curve (a much more abstract object than say, a circle). Previously, the earliest instance of this we knew of was in England in the 1300s.
If your library has access to the journal Science, you can find more details in:
Ossyndrijver, Matthieu. “Ancient Babylonian astronomers calculated Jupiter’s position from the area under a time-velocity graph.” Science. Vol. 351, Issue 6272, pp. 482-484.