The extension to higher dimension of differentiation was established in the previous chapters. We then defined line integrals which are, in a sense, one dimensional integrals which exist in a high dimensional setting. We now take the next step and define higher dimensional integrals in the sense of how to integrate a scalar field defined on a subset of
Definition of the integral
First we need to find a definition of integrability and the integral. Then we will proceed to study the properties of this higher dimensional integral. Recall that, in the one-dimensional case integration was defined using the following steps:
Define the integral for step functions,
Define integral for "integrable functions",
Show that continuous functions are integrable.
For higher dimensions we follow the same logic. We will then show that we can evaluate higher dimensional integrals by repeated one-dimensional integration.
Observe that a partition divides
Definition (step function)
We can now define the integral of a step function in a reasonable way. The definition here is for 2D but the analogous definition holds for any dimension.
This should remind you of Riemann sums from Analysis I. Observe that the value of the integral is independent of the partition, as long as the function is constant on each sub-rectangle. In this sense the integral is well-defined (not dependent on the choice of partition used to calculate it).
All properties follow from the definition by basic calculations.
We are now in the position to define the set of integrable functions. In order to define integrability we take advantage of "upper" and "lower" integrals which "sandwich" the function we really want to integrate.
Definition (integrability on a rectangle)
for every pair of step functions
All the basic properties of the integral of step functions, as stated in the above Theorem, also hold for the integral of any integrable functions. This can be shown by considering the limiting procedure of the upper and lower integral of step functions which are part of the definition of integrability.
The most important words in the definition are "only one number": that's what we need to check to verify that a function is integrable. That still isn't immediately easy to check and so it is convenient to now investigate the integrability of continuous functions.
Continuity implies boundedness and so upper and lower integrals exist. Let
Evaluation of multiple integrals
Now we have a definition, so we know what a multidimensional integral is, and we also know that some interesting ones exist, but it is essential to also have a way to practically evaluate any given integral. It turns out we can do that by integrating in one variable at a time:
To see this, think about any pair of step functions
since these are all just different names for the same sum, and the same is true for
in other words the iterated integral in the middle is bounded from above and below by the same upper and lower integrals as the integral of
and the other equality holds for the same reason.
This integral naturally allows us to calculate the volume of a solid. Let
The volume of the set
Up until now we have considered step function and continuous functions. As with one-dimensional integrals we can permit some discontinuities and we introduce the following concept to be able to control the functions with discontinuities sufficiently to guarantee that the integrals are well-defined.
Definition (Content zero sets)
A bounded subset
Examples of content zero sets include: finite sets of points; bounded line segments; continuous paths.
By continuity, for every
Take a cover of
Regions bounded by functions
A major limitation is that we have only integrated over rectangles whereas we would like to integrate over much more general different shaped regions. This we develop now.
We use this notation in the following definition.
We say that
Suppose that there are continuous functions
Not all sets can be written in this way but many can and such a way of describing a subset of
the set of discontinuity of
We could also consider the following set
which we will call a Type 2 set. This is just the same situation as above with the roles of
In the first case we could describe the representation as projecting along the
For higher dimensions we need to also have an understanding of how to represent subsets of
In order to describe this set it is convenient to imagine how it projects down onto the