Chapter 17: Double and Triple Integrals
      Section 17.1 Multiple Sigma Notation                                           Section 17.5 Further Applications of the Double Integration
      a.  Double Sigma Notation                                                      a.   Mass of a Plate
      b.  Properties                                                                 b.   Center of Mass of a Plate
                                                                                     c.   Centroids
      Section 17.2 Double Integrals                                                  d.   Applications
      a.  Double Integral Over a Rectangle
      b.  Partitions                                                                 Section 17.6 Triple Integrals
      c.  More on Partitions                                                         a.   Triple Integral Over a Box
      d.  Upper Sums and Lower Sums                                                  b.   Triple Integral Over a More General Solid
      e.  Double Integral Over a Rectangle R                                         c.   Volume
      f.  Double Integral as a Volume
      g.  Volume of T                                                                Section 17.7 Reduction to Repeated Integrals
      h.  Double Integral Over a Region                                              a.   Formula and Illustration
      i.  Volume of the Solid T
      j.  Elementary Properties: I and II                                            Section 17.8 Cylindrical Coordinates
      k.  Elementary Property III                                                    a.   Rectangular Coordinates/ Cylindrical Coordinates
      l.  Elementary Property IV                                                     b.   Evaluating Triple Integrals
      m. Mean-Value Theorem for Double Integrals                                     c.   Volume in Cylindrical Coordinates 
      Section 17.3 The Evaluation of Double Integrals By Repeated Integrals          Section 17.9 Spherical Coordinates
      a.  Type I Regions                                                             a.   Longitude, Colatitude, Latitude
      b.  Type II Region                                                             b.   Spherical Wedge
      c.  Reduction Formulas Viewed Geometrically                                    c.   Volume
      d.  Reduction Formula
      e.  Symmetry in Double Integration                                             Section 17.10 Jacobians, Changing Variables
                                                                                     a. Change of Variables
      Section 17.4 The Double Integral as a Limit of Riemann Sums; Polar             b. Jacobian
      Coordinates
      a.  Limit of Riemann Sums
      b.  Evaluating Double Integrals Using Polar Coordinates
      c.  Double Integral Formulas
                       −x2
      d.  Integrating e                                                                                                  Salas, Hille, Etgen Calculus: One and Several Variables
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                                  Multiple Sigma Notation
            When two indices are involved, say,
                                            ij               2i                      j
                                          25,                     ,            1
                                    a=               a=                 ai= +
                                     ij                ij  5+ j           ij  (     )
            we use double-sigma notation. By
            we mean the sum of all the a where i ranges from 1 to m and j ranges from 1 
            to n. For example,                ij
                  32ij                          22 223 32
                 ∑∑25=2⋅52+⋅5+2⋅52+ ⋅5+2⋅+52⋅5=420
                  ij=11=
                                                                                               Salas, Hille, Etgen Calculus: One and Several Variables
                                                        Main Menu                            Copyright 2007 © John Wiley & Sons, Inc.  All rights reserved.
                                  Multiple Sigma Notation
                                                                                               Salas, Hille, Etgen Calculus: One and Several Variables
                                                        Main Menu                            Copyright 2007 © John Wiley & Sons, Inc.  All rights reserved.
                                           Double Integrals
             The Double Integral over a Rectangle
             We start with a function f continuous on a rectangle
                                               R : a ≤ x ≤ b, c ≤ y ≤ d
             We want to define the double integral of f over R:
                                                  f  x,.y dxdy
                                               ∫∫   (     )
                                               R
                                                                                               Salas, Hille, Etgen Calculus: One and Several Variables
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