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Unformatted text preview: Higher Dimensional Integration
Jed Keesling
So far we have only considered integration of a function on the real line. It is easy
enough to generalize this to higher dimensions. However, you need to be alerted to some
caveats. This will be a topic that will be covered more thoroughly in MAC 2313 or MAC
3474.
Consider the function ! ! , ! = ! ! + ! ! . We would like to determine the volume that is
under the graph of this function above the xy–plane over the square −3,3 ×[−3,3]. The
graph is given below We break up the sides of the square into small intervals and this breaks up the square into
a grid of rectangles. We approximate the volume over each of these rectangles by the
formula ∆! = !(!! , !! ) ∙ ∆! ∆! where ∆! and ∆! are the sides of the small rectangle and
the point (!! , !! ) is a point in this rectangle. We express the limit of the sums of these
small volumes as an integral. In the above example it would be given in the following
form.
!
!
!!
!! !! + ! ! !"!# We can compute this area by integrating first with respect to the variable x. We do this
by treating y as a constant. Then we integrate with respect to y. The calculation is below.
!
!
!!
!! !! = + ! ! !"!# =
! !"
!! ! ! !!
!! ! + !" ! + 6! ! !" = 216 So, we have determined the volume of the following figure. !
!! !" 2
0 2 15 10 5 0 2
0
2 We get the same result if we change the order of integration and integrate first with
respect to y holding x constant and then integrate with respect to x.
For continuous functions on bounded domains, we can exchange the order of integration
with confidence. However, for unbounded domains and or unbounded functions,
changing the order of integration may give different results. Of course, if this happens,
we do not have a welldefined value for the volume. Here are two examples to show this.
The first uses the function ! ! , ! = ! ! !! ! ! ! !! ! ! . The region over which the integration takes place is 1, ∞ ×[1, ∞). Computing the iterated integration in both orders is simple.
Obviously you get quite different answers depending on the order of the integration. Here is the graph of this function over 1,10 ×[1,10]. Here is a more complicated example that we presented in class. Even though it is more
difficult to explain, it reveals more clearly the basic reason for the difference in the order
of integration. Here is a brief description of the example.
( ! ! ) !!! Consider the following series ln 2 = !!! ! . We showed in class that the terms
!
in this series could be rearranged to converge to a different number, in fact, to any other
!
number that we please. Let the terms in this series be given by !! = 1, !! = − ! , !! =
! ! , !! = − ! , …. Let us suppose that we have a rearrangement that converges to the
number 3 rather than ln(2). Suppose that the subscripts in this rearrangement are given
by !, !! , !! , …. That is, 3 = !!! !!! . We now define a function ! ! , ! on 0, ∞ ×
!
[0, ∞) in the following manner. We break up 0, ∞ ×[0, ∞) into an integer grid. We
show a portion of that grid below. Our function will be constant on these squares. It will
be zero everywhere except for one square in each column and each row. The square
where it is not zero will have coordinates (!, !) where ! = !! and on that square it will
take on the value !! . We illustrate below by showing where the function takes on the
value !! . We are assuming that !! appears as the 5th term in the rearranged sequence
whose limit is 3. That is, 1 = !! . Everywhere else in the first column the function is
zero.
! !! 7
6
5 !! 4
3
2
1
1 2 3 4 5 6 7 Now consider the two orders of integration with this example.
!!
!
!! ! , ! !"!# = !
! ! ! !! !!
!
!! ! , ! !"!# = !
! ! ! !! ! =
=3 !!! ! (! ! )
! !!
! = ln(2) Simply put, changing the order of integration changes the order in which we add the
numbers !! !!! that are the terms in the series. Integrating in the order !"!# we sum
!
the numbers in the order that gives ln(2) as the limit. Integrating in the order !"!#
we get the order of the sum that gives the number 3. ...
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This note was uploaded on 07/14/2011 for the course MAC 3473 taught by Professor Block during the Spring '08 term at University of Florida.
 Spring '08
 BLOCK
 Calculus

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