# Multiple integrals

See also the graded exercises and additional exercises associated to this part of the course. If you want more, Chapter 5 (plus sections 6.3 and 6.4) of OpenStax Calculus Volume 3 is a good option.

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.

Definition (partition)

Let *partition* of

Observe that a partition divides

Definition (step function)

A function *step function* if there is a partition

If

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.

Suppose that

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).

Theorem

Let

Proof

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)

Let *integrable* function if there is one and only one number

for every pair of step functions

This number

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.

Theorem

Suppose that

Proof

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:

Theorem (Fubini)

Let

Proof

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.

Theorem

Suppose that

Proof

By continuity, for every

Theorem

Let

Proof

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.

Suppose

We use this notation in the following definition.

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

Let

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

where