I am familiar with the idea of a LINE INTEGRAL of a SCALAR QUANTITY, defined in (say) three-dimensional space, with respect to the arc length;
and this generally also results in a scalar quantity.
Stroud's Advanced Engineering Mathematics, in Programme 18, evaluates
this to a VECTOR - which I have not seen elsewhere - and gives no
indication as to what this really means.
I THINK the length of this vector is the quantity which would result
from the integration w.r.t. the arc. Stroud simply leaves the vector
as "the answer". I would be grateful for confirmation that another step is
Does the vector have an interpretation or is it just an intermediate result?