MA2321—Analysis in Several Variables
     School of Mathematics, Trinity College
          Michaelmas Term 2018
         Section 6: Multiple Integrals
              David R. Wilkins
  6. Multiple Integrals
                6. Multiple Integrals
     6.1. Multiple Integrals of Bounded Continuous Functions
    Weconsiders integrals of continuous real-valued functions of
    several real variables over regions that are products of closed
    bounded intervals. Any subset of n-dimensional Euclidean
    space Rn that is a product of closed bounded intervals is a closed
              n
    bounded set in R . It follows from the Extreme Value Theorem
    (Theorem 4.21) that any continuous real-valued function on a
    product of closed bounded intervals is necessarily bounded on that
    product of intervals. It is also uniformly continuous on that
    product of intervals (see Theorem 4.22)
  6. Multiple Integrals (continued)
       Proposition 6.1
       Let n be an integer greater than 1, let a ,a ,...,a and
                                                1  2       n
       b ,b ,...,b be real numbers, where a < b for i = 1,2,...,n, let
        1  2       n                           i    i
       f : [a ,b ] × ··· × [a ,b ] → R be a continuous real-valued
            1  1            n  n
       function, and let
               g(x ,x ,...,x     ) = Z bn f (x ,x ,...,x    , t) dt.
                   1  2       n−1      a      1  2       n−1
                                        n
       for all (n − 1)-tuples (x1,x2,...,xn−1) of real numbers satisfying
       a ≤x ≤b for i = 1,2,...,n−1. Then the function
        i    i    i
                   g: [a ,b ] × [a ,b ]··· × [a    , b   ] → R
                        1  1      2  2         n−1   n−1
       is continuous.
  6. Multiple Integrals (continued)
       Proof
       Let some positive real number ε be given, and let ε be chosen so
                                                             0
       that 0 < (b −a )ε < ε. The function f is uniformly continuous
                   n     n  0
       on [a ,b ] × [a ,b ]··· × [a ,b ] (see Theorem 4.22). Therefore
            1   1      2  2          n  n
       there exists some positive real number δ such that
                |f (x ,x ,...,x    , t) − f (u ,u ,...,u    , t)| < ε
                    1  2       n−1           1  2       n−1         0
       for all real numbers x ,x ,...,x     , u ,u ,...,u      and t
                              1  2       n−1   1   2       n−1
       satisfying a ≤ x ≤ b , a ≤ u < b and |x −u | < δ for
                   i    i     i  i     i    i       i    i
       i = 1,2,...,n −1 and a ≤ t ≤ b . Applying Proposition 5.7, we
                                 n         n
       see that