How to integrate 2 compoonent spline function

Question

I have a Spline function defined as follows.

Points = {{0, 0}, {0, 1}, {1, 1}, {1, 0}, {0, 0}};

bsf = BSplineFunction[Points, SplineDegree -> 1];

A Parametric Plot look like this:

ParametricPlot[bsf[r], {r, 0, 1}]

But when I try to do NIntegrate, I get for the first component

NIntegrate[bsf[t][[1]], {t, 0, 1}] 

An output of 0.5 as it should be, but for the second component

NIntegrate[bsf[t][[2]], {t, 0, 1}]

I get as an output

I still get the correct output though. But If I just type

NIntegrate[bsf[t], {t, 0, 1}]

I just get error

Why is this and how can I fix this?

Thanks in advance!

Answer 1

Indexed.

NIntegrate[Indexed[bsf[t], 1], {t, 0, 1}]
NIntegrate[Indexed[bsf[t], 2], {t, 0, 1}]

0.5.

0.5.

Answer 2

There are two options I could come up with here. The first is to define your bsf function to only accept numerical inputs. That is,

Points = {{0, 0}, {0, 1}, {1, 1}, {1, 0}, {0, 0}};
bsf = BSplineFunction[Points, SplineDegree -> 1];
bsfNum[t_?NumericQ, index_] := bsf[t][[index]]

Then,

NIntegrate[gg[t, 1], {t, 0, 1}]
NIntegrate[gg[t, 2], {t, 0, 1}]
(* 0.5 *)
(* 0.5 *)

The issue is that bsf[t] doesn't evaluate to a list of two elements until after you have entered a numerical value for t, and this forces Mathematica to define a numerical function instead.

Alternatively, we can construct the B-spline functions from the BSplineBasis functions, using the knots generated internally by BSplineFunction. We define,

bsfKnot[t_] = Sum[
   Points[[j + 1]] BSplineBasis[{1, Rationalize@First@bsf["Knots"]}, j, t],
   {j, 0, Length@Points - 1}
  ];

Then, this seems to be the right function:

ParametricPlot[bsfKnot[t], {t, 0, 1}]

In addition, we can use Integrate or NIntegrate:

Integrate[bsfKnot[t][[2]], {t, 0, 1}]
NIntegrate[bsfKnot[t][[2]], {t, 0, 1}]
(* 1/2 *)
(* 0.5 *)