Integrate function returns vector instead of scalar

Question

When I try to integrate a function mapping a vector to a scalar Mathematica "ignores" the function and returns a vector independent of the function.

Here is a minimal example of my problem: I want to integrate the sum over all absolute values in a vector over a 2D unit circle. When I use the Total function for this, i.e. f1 Mathematica returns a scalar (2.67) that looks reasonable. When I use my f2 function which - as far as I know - should basically implement the Total function, it does not return a scalar but a vector {0., 0.}.

f1[x_] := Total[x]
f2[x_] := Apply[Plus, x]
NIntegrate[f1[Abs[x]], x \[Element] Ball[ConstantArray[0, 2]]]
NIntegrate[f2[Abs[x]], x \[Element] Ball[ConstantArray[0, 2]]]

When I directly compare what my f2 function does it looks like it behaves like the f1 function, i.e. f1[{0.2, 0.3 }] == f2[{0.2, 0.3}] returns True.

I am puzzled by the (seemingly) different behavior of Mathematica depending on whether I use the f2 function inside NIntegrate or outside; now I am looking for an explanation what happens here and how and why I have to modify the f2 function such that it behaves like f1 in my integral.

Answer 1

The problem seems to be because f2[Abs[x]] evaluates to x symbolically:

ClearAll[f1, f2]; (* be sure to clear previous definitions *)
f1[x_] := Total[x]
f2[x_?VectorQ] := Apply[Plus, x]
NIntegrate[f1[Abs[x]], x \[Element] Ball[ConstantArray[0, 2]]]
NIntegrate[f2[Abs[x]], x \[Element] Ball[ConstantArray[0, 2]]]

(*
2.66667
2.66667
*)

Answer 2

Don't know why Mathematica doesn't recognize xto be a list!

Try

NIntegrate[f1[Abs[{x1, x2}]], {x1, x2} \[Element]Ball[ConstantArray[0, 2]]]
(*2.66667*)

NIntegrate[f2[Abs[{x1, x2}]], {x1, x2} \[Element]Ball[ConstantArray[0, 2]]]    
(*2.66667*)