For example, if your stringline is 40 feet long when it spans the depression, and it is also 40 feet wide in the cross direction (circular), and the depth is 6 inches, then you will need:

40 X 40 X 6 /800 = 12 cubic yards

If the depression is elliptical, e.g., 40 feet long, 20 feet wide, and 4 inches deep:

40 X 20 X 4/800 = 4 cubic yards

The shape of the depression makes little difference, as long as it all goes down to one center. For example, a circular depression 40 feet across with a conical shape and a flat bottom 20 feet across will need 13.6 cubic yards, compared with 12 cubic yards above. In the extreme, a depression, with a perfectly flat bottom and perfectly vertical sides will need 23.27 cubic yards.

These calculations can be verified by the formulae for partially filled spheres and cones: http://grapevine.abe.msstate.edu/~fto/tools/vol/

Getting further from pure geometry, depressions can undulate, with some ups and downs, which will require less backfill. In those cases it would be better to isolate the individual deepest depressions for measurements, and add up the results for the separate depressions.

Just curious - wouldn't it be easier to move the dirt from the high points to the low points?

The thing is; the high points are the original height of the yard and follow the irrigation intentions. When the town decided to do some road work, they dug up some of the yard and when they allegidly repaired it, they left a sort of rough terrain ditch in the middle of the yard. I think they took dirt with them. The ditch they they have made is more like a rectangular lake when it rains. That is, it tapers, it is not just an immediate drop off. The water valve used to be flush with the lawn now it sits about eighteen inches above it.