Extensions

You could also estimate the area with Simpson's Rule

If you have sensitive scales available, another interesting and quite different way to approximate the area is to:
1. Cut out the entire grid and weigh it.
2. Compute g/unit^2 mass density for the whole grid.
3. Carefully, cut out the region and weigh it.
4. Use the results from 2 and 3 to approximate the area.

If you try this, it might work better if you can get the region copied onto heavy paper or poster board.