Let $D$ be the area bounded by a series of points $(x_i,y_i)_{i=1}^{N}$.(The area need not to be convex and the points are supposed to go along the boundary curve.)
Let $f$ be a function defined on $D$ but we only know its values on a given point set (finite and discrete), say (x'_i,y'_i,f(x'_i,y'_i))_{i=1}^{N'}.(The given data set need not to be "dense" in $D$.)
How can I do numeric integration of $f$ over $D$?
Here is what I think:
1) First we should approximate the boundary of $D$ by segments between those series of points.
2) Then we should do some interpolation on the given data set. However, interpolation in two-dimension is not always possible. Then I get stuck.
Can you please help? Thank you.
EDIT: The function value are only known on ($(x'_i,y'_i)$ in the interior of $D$. Its values on the boundary of $D$ (where the points are $(x_i,y_i)$, without prime) are not known.