Lec16[1]Integrales Linea

8/13/2019 Lec16[1]Integrales Linea

1/35

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


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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4/35

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, 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.


5/35

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, 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.


6/35

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

.


7/35

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

.


8/35

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.


9/35

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].


10/35

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

.


11/35

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.


12/35

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.


13/35

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].


14/35

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.


15/35

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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17/35

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].


18/35

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).


19/35

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.


20/35

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 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...


21/35

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 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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23/35

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.


24/35

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


25/35

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


26/35

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 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


27/35

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 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


28/35

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 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


29/35

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 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


30/35

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 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


31/35

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 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


32/35

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


33/35

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


34/35

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


35/35

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.

Lec16[1]Integrales Linea

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