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

27.4 Evaluation of line integrals in three dimensions 873
integral entirely in terms of the variablexwhich is then integrated betweenx = 2 and
x = 3.Thus
∫
C
(3x 2 +y)dx+(5x−y)dy=
=
∫ 3
2
∫ 3
2
(3x 2 +2x 2 )dx + (5x −2x 2 )(4xdx)
25x 2 −8x 3 dx
[ ] 25x
3 3
= −2x 4
3
2
= 28.333
EXERCISES27.3
1 Evaluate ∫ C3ydx +2xdyalong the straightlineC
between(1,1)and(3,3).
2 Evaluate ∫ C 2yxdx +x2 dyalong the straightline
y = 4xfrom (0,0)to (3,12).
3 Evaluate ∫ C (7x +3y)dx +2ydy alongthe curve
y =x 2 between (0,0)and (2,4).
4 IfE = (x +2y)i + (x −3y)j,Aisthepoint (0,0)and
Bis the point (3,2),evaluate
∫ B
E·ds
A
(a)along the straightline joining AandB,
(b)horizontally along thexaxisfromx = 0 tox = 3
andthenverticallyfromy = 0 toy = 2.
5 IfF
∫
= (2xy −y 4 +3)i + (x 2 −4xy 3 )jevaluate
CF · dswhereC is the straightlinejoining A(1,3)
andB(2,5).
6 Evaluate the integral
∫
(3x 2 +2y)dx + (7x +y 2 )dy
C
from A(0,1)to B(2, 5)along the curveC defined by
y=2x+1.
Solutions
1 20
2 108
3 38
4 (a)
15
2
5 −1149
6
268
3
(b)
9
2
27.4 EVALUATIONOFLINEINTEGRALSINTHREEDIMENSIONS
Evaluation of line integrals along curves lying in three-dimensional space is performed
inasimilar way. It isoften helpful inthis case toexpress the independentvariablesx,y
andzintermsofasingleparameter. Consider the following example.