Here you go:
1) Do not draw the axis
2) Do not draw the plots with * but plot it with - .
3) Also plot $y=-0.02$, $x=-0.02$, $x=0.08$ plots. Plot the upper boundary as well.
4) Now take a screen capture. The bigger the plot, the more accurate the result will be.
5) Normalize the image to $0-1$. (Make it logical as in $logical(I)$)
6) Apply connected component labeling (supposedly in matlab using bwlabel etc.) - If you are not familiar with this part, I could elaborate.
Now you have closed shapes as connected binary regions. You can compute the region properties with : http://www.mathworks.com/help/images/ref/regionprops.html
Center and area are just 2 properties of these regions.
7) If you want to paint those regions on the image you could as well do so, since you have every point in the region.
If you don't have the image processing toolbox, then let me put it this way (exactly the same operation):
Sufficiently discretize your grid (adjust grid resolution so that the accuracy is sufficient for you). Each discrete point (a.k.a pixel) will be a node in your grid. Then for all grid locations other than the actual data points, form a connectivity graph (adjacency graph). This will connect the neighboring nodes (Use 4 or 8 connectivity) Then apply Tarjan's strongly connected components analysis to it: http://en.wikipedia.org/wiki/Tarjan's_strongly_connected_components_algorithm .
At this point, you will end up a component label per each node. This is the same as obtaining the connected components from the image. After that, you could retrieve each discretized point on the grid separately to compute the centroid, area or you can visualize these by label coloring.
Hope this helps.