Consider a vector function #~v #: [ ~a , ~b ] -> V_3 , _ i.e. #~v ( ~t ) &in. V_3 , ~t &in. [ ~a , ~b ]. _ Let
#~v_~n _ = _ sum{#~v ( ~t_~r ) &delta._~n~t,~r = 0,~n - 1}
where _ &delta._~n~t = ( ~b - ~a ) ./ ~n , _ and _ ~t_~r = ~a + ~r &delta._~n~t. _ Now _ &delta._~n~t -> 0 as ~n -> &infty.. define the #~{definite integral} of #~v over [ ~a , ~b ] as
int{#~v (~t) d~t,~a,~b,} _ #:= _ lim{#~v_~n,~n -> &infty.}
if #~v is expressed in terms of its coordinate functions:
#~v (~t) _ = _ #~i ~v_~x (~t) + #~j ~v_~y (~t) + #~k ~v_~z (~t)
then
int{#~v (~t) d~t,~a,~b,} _ #:= _ #~i int{~v_~x (~t) d~t,~a,~b,} + _ #~j int{~v_~y (~t) d~t,~a,~b,} + _ #~k int{~v_~z (~t) d~t,~a,~b,}