8.4 – exercأچcios – pg. 344 ( ) 3 12 3 3 3 1 1 1 1 2 3 3 0 3 3 2 3 0 2 3 0 2 4 3 0 2...

Download 8.4 – EXERCأچCIOS – pg. 344 ( ) 3 12 3 3 3 1 1 1 1 2 3 3 0 3 3 2 3 0 2 3 0 2 4 3 0 2 2 = + = + =

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  • 8.4 – EXERCÍCIOS – pg. 344

    Nos exercícios de 1 a 14, encontrar o comprimento de arco da curva dada.

    1. 25 −= xy , 22 ≤≤− x

    ( )

    ( ) ..2642226

    262651

    1

    2

    2

    2

    2

    2

    2

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    cu

    xdxdx

    dxxfs

    b

    a

    =+=

    ==+=

    ′+=

    −−−

    ∫∫

    2. 13 2

    −= xy , 21 ≤≤ x

    3 1

    3

    2 − =′ xy

    dxxdxx

    dx

    x

    x dx

    x

    s

    3 1

    2

    1

    2 1

    3 2

    2

    1 3

    2

    3 22

    1 3

    2

    . 3

    1 .49

    9

    49

    9

    4 1

    ∫∫

      

      +=

    + =+=

     

     

     −

     

      +=

     

     

     −

     

      +=

      

      +

    =

      

      +=

    31342.9 27

    1 1349

    3

    2 .

    18

    1

    2 3

    49

    . 18

    1

    .6.49 6

    1 .

    3

    1

    2 3

    3 2

    2 32

    3

    3 2

    2

    1

    2 3

    3 2

    3 1

    2

    1

    2 1

    3 2

    x

    x

    dxxx

    3. ( ) 2 3

    22 3

    1 xy += , 30 ≤≤ x

    ( ) xxy 2.2 2

    3 .

    3

    1 2

    1 2+=′

  • ( )

    ( )

    ( ) 123 3

    3

    3 1

    1

    1

    21

    3 3

    0

    33

    0

    2

    3

    0

    2

    3

    0

    42

    3

    0

    22

    =+=+=+=

    +=

    ++=

    ++=

    x x

    dxx

    dxx

    dxxx

    dxxxs

    4. 3 2

    3 2

    3 2

    2=+ yx

    

      

    =

    =

    tseny

    tx

    3

    3

    2

    cos2

    ( )

    ..12

    2 .24cos.24

    coscos.64

    cos36cos364

    2

    0

    22

    0

    2

    0

    2222

    2

    0

    2424

    cu

    tsen dtttsen

    dttsentttsen

    dtttsentsents

    =

    ==

    +=

    +=

    ππ

    π

    π

    5. 2

    4

    8

    1

    4

    1

    x xy += , 21 ≤≤ x

    ( ) 33 2 8

    1

    4

    1 −−+=′ xxy

  • ( ) dxx x

    dx x

    xx

    dx x

    xx

    dx xx

    xx

    dx x

    xs

    +=

    ++ =

    −+=

    +−+=

     

      

     −+=

    2

    1

    26

    3

    2

    1

    6

    126

    2

    1

    6

    36

    2

    1

    3

    36

    2

    1

    2

    3

    3

    14 4

    1

    16

    1168

    16

    1 .21

    616

    1

    4

    1 .21

    4

    1 1

    ( )

    ( )

    32

    123

    2

    1 1

    2.2

    1 2

    4

    1

    24 .4

    4

    1

    4 4

    1

    14 4

    1

    2

    4

    2

    1

    24

    2

    1

    33

    2

    1

    26

    3

    =

     

      

     +−−=

     

      

     +=

    +=

    +=

    xx

    dxxx

    dxx x

    6. y

    yx 4

    1

    3

    1 3 += , 31 ≤≤ y

    ( )

    2

    4

    2

    2

    22

    4

    14

    4

    1

    .1 4

    1 3.

    3

    1

    y

    y

    y y

    yyx

    − =

    −=

    −+=′ −

  • ( ) 4

    24

    4

    48

    4

    484

    4

    48 2

    2

    4

    16

    14

    16

    1816

    16

    181616

    16

    1816 1

    4

    14 1

    y

    y

    y

    yy

    y

    yyy

    y

    yy

    y

    y

    − =

    ++ =

    +−+ =

    +− +=

      

     − +

    ( )

    6

    53

    4

    1

    3

    1

    3.4

    1

    3

    3

    1 .

    4

    1

    3

    4

    1

    4

    14

    16

    14

    3

    3

    1

    13

    3

    1

    22

    3

    1

    2

    4

    3

    1

    4

    24

    =

    +−−=

     

      

    − +=

     

      

     +=

    + =

    + =

    ∫∫

    yy

    dyyydy y

    y

    dy y

    y s

    7. ( )xx eey −+= 2

    1 de ( )1,0 a 

      

     + −

    2 ,1

    1ee

    ( )xx eey −−=′ 2

    1

    ( )

    ( )

    ( )

    dx ee

    dxee

    dxee

    dxeeee

    dxees

    xx

    xx

    xx

    xxxx

    xx

    −−

    +−+ =

    +−+=

    +−+=

    +−+=

    −+=

    1

    0

    22

    1

    0

    22

    1

    0

    22

    1

    0

    22

    1

    0

    2

    4

    24

    4

    1

    2

    1

    4

    1 1

    2 4

    1 1

    ..2 4

    1 1

    4

    1 1

  • ( )

    ( )

    ( )

    ( )

    ( )

    1

    11 2

    1

    2

    1

    2

    1

    1 1

    2

    1

    1 1

    2

    1

    1 2

    1

    1

    0

    1

    0

    1

    0

    2

    1

    0

    22

    1

    0

    2

    2

    hsen

    ee

    ee

    dxee

    dxe e

    dxe e

    dx e

    e

    xx

    xx

    x

    x

    x

    x

    x

    x

    =

    +−−=

    +=

    +=

    +=

    +=

    ++=

    8. xy ln= , 83 ≤≤ x

    x y

    1 =′

    dx x

    x dx

    x s ∫∫

    + =+=

    8

    3

    28

    3

    2

    11 1

    ( )

    ( ) dtttdx

    tx

    tx

    tx

    2.1 2

    1

    1

    1

    1

    2 1

    2

    2 1

    2

    22

    22

    −=

    −=

    −=

    =+

  • ( ) ( )

    ( ) ( )∫ ∫

    ∫∫

    +− +=

     

      

    − +=

    − =

    −−

    = +

    =

    11

    1

    1 1

    1

    1

    .

    1

    1

    2

    2

    2

    2 1

    22 1

    2

    2

    tt

    dt dt

    dt t

    t

    dtt

    t

    dtt

    t

    t dx

    x

    x I

    C x

    x x

    C t

    t t

    Cttt

    dt t

    dt t

    t

    + ++

    −+ ++=

    + +

    − +=

    ++−−+=

    + −

    − += ∫∫

    11

    11 ln

    2

    1 1

    1

    1 ln

    2

    1

    1ln 2

    1 1ln

    2

    1

    1

    2 1

    1

    2 1

    2

    2 2

    2

    3 ln

    2

    1 1

    3

    1 ln

    2

    1 2

    4

    2 ln

    2

    1 3

    11

    11 ln

    2

    1 1

    8

    3

    2

    2 2

    +=

     

      

     −−+=

     

     

    ++

    −+ ++=

    x

    x xs

    9. ( )xseny ln1−= , 46

    ππ ≤≤ x

    xsen

    x y

    cos −=′

  • ∫∫

    =

    = +

    =

    +=

    4

    6

    4

    6

    4

    6

    2

    22

    4

    6

    2

    2

    cos

    cos

    cos 1

    π

    π

    π

    π

    π

    π

    π

    π

    dxxec

    sen

    dx dx

    xsen

    xxsen

    dx xsen

    x s

    x

    .. 332

    12 ln

    262

    632 ln

    132

    3 .

    2

    22 ln

    3

    3 2ln1

    2

    2 ln

    cotcosln 4

    6

    cu

    xgec

    − =

    − =

    − =

    −−−=

    −= π

    π

    10. 3xy = , de ( )0,00P ate ( )8,41P

    2 1

    2

    3 xy =′

    ( ) ..

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