Course:(Accelerated(Engineering(Calculus(II(
Instructor:(Michael(Medvinsky(
(
(
6.2.1.1 Other*types*of*line*integral*
Line(integrals(often(used(with(line(segments((but(not(only),(therefore(we(recall(the(formula(of(

the(line(segment(that(starts(at(point( r0 (and(ends(at(point r1 :( r ( t ) = (1− t ) r0 + tr1 (where( 0 ≤ t ≤ 1 .(
xy
Ex 5. Evaluate ∫
ds where C is the line from r0 = 1,2 to r1 = 3,−1 : The
13
C

parameterization curve is r ( t ) = (1− t ) 1,2 + t 3,−1 = 1− t + 3t,2 − 2t − t = 1+ 2t,2 − 3t
1
xy
(1+ 2t )( 2 − 3t ) 4 + 9 dt = 1 2 + t − 6t 2 dt = ⎛ 2t − t 2 − 2t 3 ⎞ = 1 (
ds
=
C∫ 13 ∫0
∫0
⎜⎝
⎟⎠
2
2
13
0
xy
Ex 6. Evaluate ∫
ds where C is the line from r0 = 3,−1 to r1 = 1,2 : The
13
C

parameterization curve is r ( t ) = (1− t ) 3,−1 + t 1,2 = 3 − 3t + t,t − 1+ 2t = 3 − 2t,−1+ 3t
1
1
1
⎛
⎞
3 − 2t ) ( −1+ 3t )
xy
t2
1
(
ds = ∫
4 + 9 dt = ∫ −3 + 11t − 6t 2 dt = ⎜ −3t + 11 − 2t 3 ⎟ = (
2
⎝
⎠0 2
13
13
0
0
1
∫
r1→r0
(
Thm:( 
∫ f ( x, y ) ds =
C
∫ f ( x, y ) ds 0where(C(and(–C(is(the(same(curve(with(different(direction.(
−C
Similarly( 
∫ f ( x, y, z ) ds =
C
(
∫ f ( x, y, z ) ds .(
−C
The(line(integral( 
∫ f ( x, y ) ds (is(called(a(line(integral(with(respect(to(arc(length.(There(is(also(a(
C
line(integral(with(respect(to(x( 
∫ f ( x, y ) dx = ∫ f ( x (t ), y (t )) x '(t ) dx (and(a(line(integral(with(respect(
C
C
to(y( 
∫ f ( x, y ) dy = ∫ f ( x (t ), y (t )) y'(t ) dy .(They(are(often(occur(together:((
C
C
∫ f ( x, y ) dx + ∫ g ( x, y ) dy = ∫ f ( x, y ) dx + g ( x, y ) dy (
C
Ex 7. Compute
∫
C
C
y dx + z dy + x dz where
C1 ∪C2

C1 : r (t) = 2 + t, 4t,5t , 0 ≤ t ≤ 1
1

C2 : r (t) = 3, 4,−5t , − 1 ≤ t ≤ 0
1
1
d
d
d
∫C y dx + z dy + x dz = ∫0 4t ⋅ dt (2 + t)dt + ∫0 5t ⋅ dt (4t)dt + ∫0 (2 + t)⋅ dt (5t)dt =
1
1
1
1
0
1
1
= ∫ 4t dt + ∫ 20t dt + ∫ 5(2 + t)dt = 2 + 10 + 5(2 + ) = 24
2
2
0
0
0
0
0
0
0
d
d
d
∫C y dx + z dy + x dz = −1∫ 4 ⋅ dt (3)dt + −1∫ −5t ⋅ dt (4)dt + −1∫ 3⋅ dt (−5t)dt = 0 + 0 − −1∫ 15 dt = −15 0
2
thus
(
1
1
y dx + z dy + x dz = 24 − 15 = 9 (
2
2
C1 ∪C2
∫
77(
Course:(Accelerated(Engineering(Calculus(II(
Instructor:(Michael(Medvinsky(
(
(
6.2.2 Line integrals of Vector Field
Recall:((
1) A work done by variable force f(x) in moving a particle from a to b along the xb
axis is W = ∫ f ( x ) dx (see section 6.6).
a
2) A work done by
a constant force F in moving object from point P to point Q in

space is W = F ⋅ PQ (example 6 section 9.3)
Consider(a(variable(force( F ( x, y, z ) (along(a(smooth(curve(C.(As(usually(we(divide(C(into(small(
pieces((subQarcs)(and(such(that(the(force(is(roughly(constant(on(a(subQarc(and(the(displacement(
vector(corresponding(to(a(sample(parameter( t *j ∈ ⎡⎣t j−1 ,t j ⎤⎦ is(approximately(a(tangent(vector(more(

precisely(unit(tangent(times(displacement,(i.e.( PQ ≈ Δs j T x t *j , y t *j , z t *j ,(therefore(
( ( ) ( ) ( ))
n
( ( ) ( ) ( )) ( ( ) ( ) ( ))
W ≈ ∑ F x t *j , y t *j , z t *j ⋅ T x t *j , y t *j , z t *j Δs j .(Finally(we(take( n → ∞ (and(arrive(at(
j=1
W =
∫ F ( x, y, z ) ⋅ T ( x, y, z ) ds = ∫ F ⋅ T ds .(
C
C

r '(t )

Recall:(Unit(tangent(to(the(curve( r ( t ) = x ( t ) , y ( t ) (is( T ( t ) = 
((
r '(t )

dr dr 

=
= r ' ( t ) ⇒ dr = r ' ( t ) dt (
dt dt

⎛ 
r '(t ) ⎞ 




W =
∫C F ⋅ T ds = C∫ ⎜⎝ F ( r (t )) ⋅ r '(t ) ⎟⎠ r '(t ) dt = C∫ F ( r (t )) ⋅ r '(t ) dt = C∫ F ( r (t )) ⋅ dr (t ) = C∫ F ⋅ dr (
Ex 1. Given F ( x, y, z ) = ( x, x + y, x + y + z ) ,C : r (t ) = (sint,cost,sint + cost ) ,0 ≤ t ≤ 2π evaluate ∫ F ⋅ dr
(
C
)
F ⋅ dr = sint,sint + cost,2 ( sint + cost ) ⋅ ( cost,− sint,cost − sint ) =
(
)
= sint cost − sin 2 t − sint cost + 2 cos 2 t − sin 2 t = 2cos 2 t − 3sin 2 t = (1+ cos 2t ) −
3
(1− cos 2t ) dt (
2
2π
2π
3
sin 2t 3 ⎛ sin 2t ⎞
∫C F ⋅ dr = ∫0 (1+ cos 2t ) − 2 (1− cos 2t ) dt = t + 2 − 2 ⎜⎝ t − 2 ⎟⎠ = 2π − 3π = −π
0

Thm:(Consider( F ( x, y, z ) = P ( x, y, z ) ,Q ( x, y, z ) , R ( x, y, z ) (and( r ( t ) = x ( t ) , y ( t ) , z ( t ) (then(
b
b
a
a


∫ F ⋅ dr = ∫ F ( r (t )) ⋅ r '(t ) dt = ∫ P ( x, y, z ),Q ( x, y, z ), R ( x, y, z ) ⋅ x '(t ), y'(t ), z '(t ) dt =
C
b
b
a
a
(
= ∫ P ( x, y, z ) x ' ( t ) + Q ( x, y, z ) y' ( t ) + R ( x, y, z ) z ' ( t ) dt = ∫ P ( x, y, z ) dx + Q ( x, y, z ) dy + R ( x, y, z ) dz
(
The(recent(theorem(provides(a(connection(between(line(integrals(of(vector(fields(and(the(line(
integrals(of(scalar(fields.(
(
78(