15.Multiple Integrals

  • Upload
    cain

  • View
    241

  • Download
    2

Embed Size (px)

Citation preview

  • 8/16/2019 15.Multiple Integrals

    1/93

    Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

    Chapter 15

    Multiple Integrals

  • 8/16/2019 15.Multiple Integrals

    2/93

    Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

    15.1

    Double and Iterated Integrals over 

     Rectangles

  • 8/16/2019 15.Multiple Integrals

    3/93

    Slide 15 - 3Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    4/93

    Slide 15 - 4Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    5/93

    Slide 15 - 5Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    6/93

    Slide 15 - 6Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    7/93

  • 8/16/2019 15.Multiple Integrals

    8/93

    Slide 15 - 8Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    9/93

    Slide 15 - 9Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    10/93

    Slide 15 - 10Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    11/93

    Slide 15 - 11Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    12/93

  • 8/16/2019 15.Multiple Integrals

    13/93

  • 8/16/2019 15.Multiple Integrals

    14/93

  • 8/16/2019 15.Multiple Integrals

    15/93

  • 8/16/2019 15.Multiple Integrals

    16/93

    Slide 15 - 16Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    17/93

    Slide 15 - 17Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    18/93

    Slide 15 - 18Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    19/93

    Slide 15 - 19Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    20/93

    Slide 15 - 20Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    21/93

    Slide 15 - 21Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    22/93

    Slide 15 - 22Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    23/93

    Slide 15 - 23Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    24/93

  • 8/16/2019 15.Multiple Integrals

    25/93

    Slide 15 - 25Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    26/93

    Slide 15 - 26Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    27/93

    Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

    15.3

    Area by Double Integration

  • 8/16/2019 15.Multiple Integrals

    28/93

    Slide 15 - 28Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    29/93

    Slide 15 - 29Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    30/93

    Slide 15 - 30Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    31/93

  • 8/16/2019 15.Multiple Integrals

    32/93

    Slide 15 - 32Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    33/93

    Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

    15.4

    Double Integrals in

    Polar Form

  • 8/16/2019 15.Multiple Integrals

    34/93

    Slide 15 - 34Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    35/93

    Slide 15 - 35Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    36/93

    Slide 15 - 36Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    37/93

    Slide 15 - 37Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    38/93

    Slide 15 - 38Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    39/93

    Slide 15 - 39Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    40/93

    Slide 15 - 40Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    41/93

    Slide 15 - 41Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    42/93

    Slide 15 - 42Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    43/93

    Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

    15.5

    Triple Integrals in

    Rectangular Coordinates

  • 8/16/2019 15.Multiple Integrals

    44/93

    Slide 15 - 44Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    45/93

    Slide 15 - 45Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    46/93

    Slide 15 - 46Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    47/93

    Slide 15 - 47Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    48/93

  • 8/16/2019 15.Multiple Integrals

    49/93

    Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

    15.6

    Triple Integrals in Cylindrical and

     Spherical Coordinates

  • 8/16/2019 15.Multiple Integrals

    50/93

    Slide 15 - 50Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    51/93

    Slide 15 - 51Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    52/93

    Slide 15 - 52Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    53/93

    Slide 15 - 53Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    54/93

    Slide 15 - 54Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    55/93

    Slide 15 - 55Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    56/93

  • 8/16/2019 15.Multiple Integrals

    57/93

    Slide 15 - 57Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    58/93

    Slide 15 - 58Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    59/93

    Slide 15 - 59Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    60/93

  • 8/16/2019 15.Multiple Integrals

    61/93

    Slide 15 - 61Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    62/93

    Slide 15 - 62Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    63/93

    Slide 15 - 63Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    64/93

    Slide 15 - 64Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    65/93

    Slide 15 - 65Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    66/93

    Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

    15.7

    Moments and Centers of Mass(to students)

  • 8/16/2019 15.Multiple Integrals

    67/93

    Slide 15 - 67Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    68/93

    Slide 15 - 68Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    69/93

    Slide 15 - 69Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    70/93

    Slide 15 - 70Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    71/93

  • 8/16/2019 15.Multiple Integrals

    72/93

    Slide 15 - 72Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    73/93

    Slide 15 - 73Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    74/93

  • 8/16/2019 15.Multiple Integrals

    75/93

    Slide 15 - 75Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    76/93

    Slide 15 - 76Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    77/93

    Slide 15 - 77Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    78/93

    Slide 15 - 78Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    79/93

    Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

    15.8

    Substitutions in Multiple Integrals(to students)

  • 8/16/2019 15.Multiple Integrals

    80/93

    Slide 15 - 80Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    81/93

    Slide 15 - 81Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    82/93

    Slide 15 - 82Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    83/93

    Slide 15 - 83Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    84/93

    Slide 15 - 84Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    85/93

    Slide 15 - 85Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    86/93

    Slide 15 - 86Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    87/93

    Slide 15 - 87Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    88/93

    Slide 15 - 88Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    89/93

    Slide 15 - 89Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    90/93

  • 8/16/2019 15.Multiple Integrals

    91/93

    Slide 15 - 91Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    92/93

    Slide 15 - 92Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

  • 8/16/2019 15.Multiple Integrals

    93/93