Math& 254
Prob lem Set 29
Applications of Line Integ rals
Objectives:
•
•
object with a variable density.
Use line integrals to calculate the mass of a wire-shaped
g force on an object moving along a
Use line integrals to calculate the work done by a varyin
curve.
2
Problem 01: A wire has the shape of a~
a)
+ y 2 = 4, x ~ 0.
Find a parametrization for the curve formed by the wire.
(IQ)\
?
is given by the functi onp(x , y)
b) If the linear density of the wire at the point (x, y)
mass of the wire.
p(x 'J) ·= )(
J
2
➔
(;).. ~ (,{))
- t/ COS 2/
t,)
= x 2 , find the
;J lf
1t'
~
J
(_'_T = (i-(
l/ cos ' tt)
J.,i.=
1t-
tinr I
~ogn,·tu~ .
/ r,(t\l= J.-
l~ ' (t\ dt
/)~J ' { ~) •
.fr\cf
L
1t
fcx ,())
I\ u V\') ~
-Jf
_j j
J
· ;.. d• =
8 Qlr
~ J cit . •
--Jl
- ,i,~) (B ·( ,~coJ-sc-rr)\))
r ~ COS(TT
,52...
_
we cl.cfl 1.._ huJ.
/'
Problem 02: A force field F(x,y)
x2
1
rro8fl1 tude ·
= < x 2,xy > acts on a particle that moves once around the circle
+ y 2 = 4 in the counterclockwise direction.
a) Find a parametrization for the path taken by the particle.
"rlt)-=
~
1
<. blCq;(<:)
I
0
~Si~('-)}
tt)-= \ -~ s1~ r,1,
~cos uJ ·)
b) Calculate the work done by the force field on the particle.
tlf
J
f (><'ct ) = F(X{ d ~ {(;j
l
"' { (.;, car«il \ Ji'b\' l•) (is, Ii r. ~)
0
= ( tlc.cs2·rt)
~
J(
tTT
0
-:;
l/e,,s' Cc) , 4Ch!U\ S,h{t\) •
<-.?-r,>,
(t\ ,
1
!J.C<>I
L/CQJ{(,)s;:,l, ft))
l•\)