You can right click the fitted curve and chose "Go To xxx" to active the hidden fitted curve data worksheet. And then plot a curve by the fitted data, active the graph, click Analysis-->Calculus-->Integrate from the menu, then you can see the area from the result log.
If you dont want to generate a new fitted curve plot, you can do these by Labtalk scripts. Supposed the fitted curve worksheet named 'PolyFit1', and the second column is Y column, type the following script in the command window:
integ PolyFit1_B; integ.area=;
to get the area.
If there are lots of curves, just included the script in a loop, like: