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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    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