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

874 Chapter 27 Line integrals and multiple integrals
Example27.4 Acurve,C, isdefined parametrically by
x=4 y=t 3 z=5+t
and islocated within a vector fieldF =yi +x 2 j + (z +x)k.
(a) Find the coordinates of the point P on the curve where the parameter t takes the
value 1.
(b) Find the coordinates of the point Qwhere the parametert takes the value 3.
(c) By expressing the line integral ∫ F·ds entirely in terms oft find the value of the
C
lineintegral from PtoQ.Note thatds = dxi +dyj +dzk.
Solution (a) Whent = 1,x = 4,y = 1 andz = 6,and so Phas coordinates (4, 1,6).
(b) Whent = 3,x = 4,y = 27 andz = 8,and so Qhas coordinates (4, 27, 8).
(c) To express the line integral entirely in terms of t we note that if x = 4,
dx
= 0 so that dx is also zero. If y = t 3 then dy = 3t 2 so that dy = 3t 2 dt.
dt
dt
Similarlysincez = 5 +t, dz = 1 so thatdz = dt. The line integralbecomes
dt
∫ ∫
F·ds= (yi +x 2 j + (z +x)k)·(dxi +dyj +dzk)
C
C
∫
= ydx+x 2 dy+(z+x)dz
=
=
C
∫ t=3
t=1
∫ 3
1
0 +16(3t 2 )dt+(9+t)dt
48t 2 +9+tdt
[ ] 48t
3 3
=
3 +9t+t2 2
1
( 48(3 3 )
=
3
= 438
+ (9)(3) + 32
2
) ( 48
−
3 +9+ 1 2)
EXERCISES27.4
1 You are requiredto evaluate the lineintegral
∫
CF ·ds where F is the vector field
F = (2y +3x)i + (yz +x)j +3xykand
ds = dxi +dyj +dzk.The curveC is defined
parametrically byx =t 2 ,y = 3t andz = 2t for
values oft between0and1.
(a) Findthe coordinatesofthe point A, wheret = 0.
(b) Findthe coordinatesofthe point B,wheret = 1.
(c) Byexpressing the line integralentirely in terms
oft,evaluate the lineintegral from Ato B along
the curveC.