Numerical Integration of area defined by set of coordinates?

Question

Suppose you have a general shape defined by a bunch of coordinate points that form something that looks like a circle, ellipse, or general closed curve - how do you find the area bounded by these points?

Answer 1

1. Find the convex hull of the set of points. Record down the points at the boundary.
2. Compute the area of the polygon bounded by those points.

If those points may not define a convex polygon, you need a concave hull algorithm in step 1.

Answer 2

you would typically use Monte Carlo integration or integration on the grid for multidimensional integration. you can adapt the same approach for flat surface as well.