文档介绍：Chapter Twelve
Integration
Introduction
We now turn our attention to the idea of an integral in dimensions higher than one.
Consider a real-valued function f : D ® R , where the domain D is a nice closed subset of
Euclidean n-space R n . We shall begin by seeing what we mean by the integral of f over
the set D; then later we shall see just what such an abstract thing might be good for in real
life. Mrs. Turner taught us all about the case n = 1. As it was in extending the definition
of a derivative to higher dimensions, our definition of the integral in higher dimensions will
include the definition for dimension 1 we learned in grammar school—as always, there
will be nothing to unlearn. Let us again hark back to our youth and review what we know
about the integral of f : D ® R in case D is a nice connected piece of the real line R.
First, in this context, the only nice closed pieces of R are the closed intervals; we thus
have D is a set [a,b] , where b > a . Recall that we defined a partition P of the interval to
be simply a finite subset {x0 , x1 ,K,xn } of [a,b] with a = x0 < x1 < x2 <K< xn = b.
The mesh of a partition is max{| xi - xi - 1|:i = 12, K,n} . We then defined a Riemann sum
S(P) for this partition to be a sum
n
*
S(P) = å f (xi )Dxi ,
i =1
*
where Dxi = xi - xi- 1 is simply the length of the subinterval [xi - 1, xi ] and xi is any