I am referring to the Feynman Lectures. The second volume has the "Principle of Least Action" as one of his lectures. (See after the 2nd paragraph below figure 19-6.) Although he does not explicitly say I read other sources that regard it as an area.
But I have a problem with this. It looks to me based on the dimensions of the variables that it represents a length where the action is stationary and an area for all the variations that need to be minimized.
Is it not similar to arc length in the sense that the dimension is 1 not a square and it represents a length not an area. Depending on how you treat the arc length integral will decide if it's a functional or a function for the arc length example.