082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

16.6 Integration of vectors 493
16.6 INTEGRATIONOFVECTORS
Ifavectordependsupontimet,itisoftennecessarytointegrateitwithrespecttotime.
Recall that i, j and k are constant vectors and are treated thus in any integration. Hence
the integral
∫
I = (f(t)i+g(t)j+h(t)k)dt
issimplyevaluated as three scalar integrals, and so
( ∫ ) ( ∫ ) ( ∫ )
I = f(t)dt i + g(t)dt j + h(t)dt k
Example16.15 Ifr = 3ti +t 2 j + (1 +2t)k, evaluate ∫ 1
0 rdt.
Solution
∫ 1
0
( ∫ ) (
1 ∫ ) (
1 ∫ )
1
rdt= 3tdt i + t 2 dt j + 1+2tdt k
0
[ 3t
2
=
2
0
] 1 [ ] t
3 1
i + j +
0
3
0
0
[
t +t 2 ] 1
0k = 3 2 i + 1 3 j+2k
EXERCISES16.6
1 Givenr = 3sinti−costj+(2−t)k,evaluate ∫ π
0 rdt.
2 Given v = i −3j +k,evaluate
(a)
∫ 1
vdt
0
(b)
∫ 2
vdt
0
3 Thevector, a,is defined by
a=t 2 i+e −t j+tk
Evaluate
∫ 1
(a) adt
0
(b)
∫ 3
adt
2
(c)
∫ 4
adt
1
4 Letaandbbe two three-dimensional vectors. Isthe
following true?
∫ t2
∫ t2
∫ t2
adt× bdt = a×bdt
t 1 t 1 t 1
Recallthat ×denotes the vectorproduct.
Solutions
1 6i +1.348k
2 (a) i−3j+k (b) 2i−6j+2k
3 (a)0.333i +0.632j +0.5k
4 no
(b)6.333i +0.0855j +2.5k
(c)21i +0.3496j +7.5k