Download - Lec16[1]Integrales Linea
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8/13/2019 Lec16[1]Integrales Linea
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
MATH 209Calculus, III
Volker Runde
University of Alberta
Edmonton, Fall 2011
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8/13/2019 Lec16[1]Integrales Linea
2/35
MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Line integrals, I
The settingLet Cbe a smooth curve in R2 given by the parametricequations
x=x(t), y=y(t), t
[a, b]
or by the vector equation
r(t) =x(t)i+y(t)j.
We want to integrate a function f along Cand define the lineintegral
C
f(x, y) ds.
Geometric interpretation
Iff
0, the C
f(x, y) ds is the area of the curtain with baseCand whose height above (x, y) is f(x, y).
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Line integrals, III
Theorem
For continuous f :
C
f(x, y) ds= ba
f(x(t), y(t))
dxdt
2+
dydt
2dt.
Important
The value of the integral does not depend on theparametrization ofCas long as C is traversed exactly once ast increases from a to b.
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Line integrals, IV
To remember. . .
Let s(t) be the length of the curve from r(a) to r(t). Then
ds
dt =s(t) =
dx
dt
2+
dy
dt
2,
so that
ds=
dxdt
2+
dydt
2dt.
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, I
ExampleLet Cbe the right half of the circle x2 +y2 = 16.What is
C
xy4 ds?Set
x= 4 cos t, y= 4 sin t, t
2,
2 .
Then:
C xy4 ds= 1024
2
2
cos tsin4 t16 sin2 t+ 16 cos2 t dt
= 4096
2
2
cos tsin 4tdt= 4096
11
u4 du
= 4046
5
u5u=1
u=1
=8192
5
.
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, II
Example
Let Cbe the line segment from (0, 0) to (1, 1).Evaluate
C
xy ds.Set
x=t, y=t, t [0, 1].Then:
C
xy ds= 1
0
t21 + 1 dt= 2 10
t2 dt= 23
.
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, III
Example
Let Cbe the line segment from (a, 0) to (b, 0).
EvaluateCf(x, y) ds for arbitrary continuous f.
Setx=t, y= 0, t [a, b].
Then: C
f(x, y) ds= ba
f(t, 0) dt.
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, IV
Example
Let Cconsist of the paraboloa y=x2 from (0, 0) to (1, 1)followed by the vertical line segment from (1, 1) to (1, 2).Find C2x ds.The curve C isnotsmooth, butpiecewise smooth, i.e., of theform C=C1 C2 with C1 and C2 smooth.C1: the parabola y=x
2 from (0, 0) to (1, 1):
x=t, y=t2, t [0, 1].
C2: the line segment from (1, 1) to (1, 2):
r(t) = 1, 1 +t(1, 2 1, 1) = 1, 1 +t
fort [0
,
1], i.e.,x
= 1,y
= 1 +t
, andt [0
,
1].
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, V
Example (continued)
Then:
C
2x ds=C1
2x ds+C2
2x ds
=
10
2t
1 + 4t2 dt+
10
dt
=14
5
0
u du+2 = u3
2
6
u=5
u=1
+2 =55 16
+2 =55 116
.
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Types of line integrals, I
More line integrals
We callC
f(x, y) ds the line integralwith respect to arclength.Suppose that C is given by the parametric equations
x=x(t), y=y(t), t [a, b].
Thendx
dt =x(t) and
dy
dt =y(t),
so that
dx=x(t) dt and dy=y(t) dt.
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Types of line integrals, II
DefinitionThe line integrals off swith respect to x andwith respect to y,respectively, are defined as
C f(x, y) dx :=
ba f(x(t), y(t))x
(t) dt;
and
C
f(x, y) dy :=
ba
f(x(t), y(t))y(t) dt.
Shorthand
C
P(x, y) dx+Q(x, y) dy= C
P(x, y) dx+ C
Q(x, y) dy.
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, VI
Example
Evaluate
Cy2 dx+x dy where C is the line segment from
(
5,
3) to (0, 2).
Parametrize the curve as
r(t) = 5,3 +t(0, 2 5,3) = 5 + 5t,3 + 5t,
so that
x= 5 + 5t, y= 3 + 5t, t [0, 1].
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, VII
Example (continued)
Thus:
C y
2
dx+x dy= 1
0 (5t 3)2
5 dt+ 1
0 (5t 5)5 dt= 5
10
25t2 25t+ 4 dt dt
= 5 25t3
3 25t2
2 + 4tt=1
t=0
= 56.
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, VIII
Example
EvaluateCy
2
dx+x dy where C is the arc of the parabolax= 4 y2 from (5,3) to (0, 2).The parametric equations are
x= 4 t2, y=t, t [3, 2].
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, X
Example
Evaluate
C
y2 dx+x dy where C is the line segment fromfrom (0, 2) to (
5,
3).
Parametrize the curve as
r(t) = 0, 2 +t(5,3 0, 2) = 5t, 2 5t,
so that
x= 5t, y= 5t+ 2, t [0, 1].
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, XI
Example (continued)
We obtain:
C
y2 dx+x dy 1
0
(5t+ 2)2(5) dt+ 1
0
25t dt
= 5 1
0
25t2 25t+ 4 dt
=5
6.
Note
This is precisely the negative of
Cy2 dx+x dy where C is the
line segment from (
5,
3) to (0, 2).
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Properties of line integrals
DefinitionIfC is any curve in R2, we writeCfor the curve withreversed orientation.
Properties
We have C
f(x, y) dx= C
f(x, y) dx
and C
f(x, y) dy= C
f(x, y) dy,
but C
f(x, y) ds=
C
f(x, y) ds.
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MATH 209
Calculus,III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Line integrals in R3, I
As in R2. . .
C
f(x, y, z) ds
= b
a
f(x(t), y(t), z(t))
dx
dt2
+ dy
dt2
+ dz
dt2
dt
and...
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MATH 209
Calculus,
III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Line integrals in R3, II
As in R2. . . (continued)
and...
C P(x, y, z) dx+Q(x, y, z) dy+R(x, y, z) dz
=
ba
P(x(t), y(t), z(t))x(t) dt
+ ba
Q(x(t), y(t), z(t))y(t) dt
+
ba
R(x(t), y(t), z(t))z(t) dt.
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MATH 209
Calculus,
III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, XIII
Example
Let C=C1 C2 with:
C1 = line segment from (2, 0, 0) to (3, 4, 5),
C2 = line segment from (3, 4, 5) to (3, 4, 0).
Evaluate
C
y dx+
z dy+
x dz
=
C1
y dx+z dy+x dz+
C2
y dx+z dy+x dz.
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MATH 209
Calculus,
III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, XIV
Example (continued)
Parametrize C1:
r(t) = 2, 0, 0 +t(3, 4, 5 2, 0, 0) = 2 +t, 4t, 5t,
i.e.,
x= 2 +t, y= 4t, z= 5t, t [0, 1].
Thus:
C1
y dx+ z dy+ x dz=
10
4t dt+4
10
5t dt+ 5
10
2 + t dt
= 1
0
10 + 29t dt= 10t+29t2
2 t=1
t=0
=49
2 .
E l XV
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MATH 209
Calculus,
III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, XV
Example (continued)
Parametrize C2:
x= 3, y= 4, z= 5 5t, t [0, 1].
Thus: C2
y dx+z dy+x dz= 15 1
0
dt= 15.
All in all: C
y dx+z dy+x dz=49
2 15 =19
2 .
Li i l f fi ld I
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MATH 209
Calculus,
III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Line integrals of vector fields, I
Problem
Let F=Pi + Qj + Rkbe a continuous vector field on R3 whichmoves a particle along a smooth curve C. What is the work W
done?
Easy case
IfF is constant and moves the particle along a line segmentfrom P to Q,
W =F D,where D=
PQ is the displacement vector.
Li i l f fi ld II
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MATH 209
Calculus,
III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Line integrals of vector fields, II
The general case
Divide C into n subarcs Pj1, Pjwith lengths sj by dividing
the parameter interval [a, b] into n subintervals of equal length.Choose a point Pj(x
j , y
j , z
j) on the j-th subarc, and lettj [tj1, tj] be the corresponding parameter.If sj is small: as the particle moves from Pj1 to Pj, itproceeds approximately in the direction ofT(tj), the unit
tangent vector to C at Pj(xj , y
j , z
j).
Li i t l f t fi ld III
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MATH 209
Calculus,
III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Line integrals of vector fields, III
The general case (continued)
IfWj is the work to move the particle from Pj1 to Pj, then
Wj F(xj , yj , zj) sjT(tj).
Hence,W
nj=1
F(xj , y
j , z
j) sjT(tj),
and thus
W = limn
nj=1
F(xj , y
j , z
j) sjT(tj)
= F
F(x, y, z)
T(x, y, z) ds.
Li i t l f t fi ld IV
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MATH 209
Calculus,
III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Line integrals of vector fields, IV
The general case (continued)Let Cbe given by the vector equation
r(t) =x(t)i+y(t)j+z(t)k, t [a, b].
Then:T(t) =
r(t)
|r(t)| .
Thus:
C
F T dx=
b
a
F(r(t)) r(t)|r(t)| |r(t)| dt
=
ba
F(r(t)) r(t) dt=
Line integrals of vector fields V
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MATH 209
Calculus,
III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Line integrals of vector fields, V
The general case (continued)
= b
a
F(r(t)) r(t) dt
=
ba
P(x(t), y(t), z(t))x(t) dt
+ b
a
Q(x(t), y(t), z(t))y(t) dt
+
b
a
R(x(t), y(t), z(t))z(t) dt
=
C
P(x, y, z) dx+Q(x, y, z) dy+R(x, y, z) dz.
Line integrals of vector fields VI
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MATH 209
Calculus,
III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Line integrals of vector fields, VI
Definition
Let F= Pi+Qj+Rk be a continuous vector field on R3, andlet Cbe a smooth curve given by the vector function r(t) for
t [a, b]. Theline integral ofF along C isC
F dr= ba
F(r(t)) r(t) dt
=C
P(x, y, z) dx+Q(x, y, z) dy+R(x, y, z) dz.
Examples XVI
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MATH 209
Calculus,
III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, XVI
ExampleA force field
F(x, y) =xsin yi+yj
moves a particle from (
1, 1) to (2, 4) along the parabola
y=x2. Compute the total work W.Parametrize C as
r(t) =ti+t2j, t [1, 2].
Then:r(t) =i+ 2tj
andF(r(t)) =tsin(t2)i+t2j.
Examples XVII
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MATH 209
Calculus,
III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, XVII
Example (continued)
Thus:
W =C
F dr= 21
(tsin(t2)i+t2j) (i+ 2tj) dt
=
21
tsin(t2) + 2t3 dt=1
2
41
sin u du+ t4
2
t=2
t=1
=cos u2
u=4u=1
+152
=12
(15 cos 4 + cos 1).
Examples XVIII
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MATH 209
Calculus,
III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, XVIII
ExampleLet
F(x, y, z) =xyi+yzj+zxk,
and let Cbe given by
x=t, y=t2, z=t3, t [0, 1].
i.e., byr(t) =ti+t2j+t3k, t [0, 1].
Thus:r(t) =i+ 2tj+ 3t2k
andF(r(t)) =t3i+t5j+t4k.
Examples XIX
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MATH 209
Calculus,
III
Volker Runde
Line integrals
in R2
Types of line
integrals
Line integrals
in R3
Line integrals
of vector fields
Examples, XIX
Example (continued)
Therefore:
C
F
dr= 10
t3 + 2t6 + 3t6 dt
=
10
t3 + 5t6 dt
= t4
4 +
5t7
7t=1
t=0
=27
28.