This chapter is on the Riemannn integral for a function of n variables. It begins by introducing the basic concepts and applications of the integral. The general considerations including the definition of the integral and proofs of theorems are left till later. These are very difficult topics and are likely better considered in the context of the Lebesgue integral. Consider the following region which is labeled R.
We will consider the following iterated integral which makes sense for any continuous function f

It means just exactly what the notation suggests it does. You fix x and then you do the inside integral

This yields a function of x which will end up being continuous. You then do ∫ _{a}^{b}dx to this continuous function.
What was it about the above region which made it possible to set up such an iterated integral? It was just this: You have a curve on the top y = t
Example 24.1.1 Suppose t
You should sketch the graphs of these functions. Filling in the limits as above, we obtain

Of course one could do the iterated integral in the other order for this example. In this case, you would be considering a curve on the left x = −

Why should it be the case that these two iterated integrals are equal? This involves a consideration of what you are computing when you do such an iterated integral. First note that in the general example given above involving t
For simplicity, we let the distance between the vertical lines be Δx and the distance between the horizontal lines be Δy. We will only consider those rectangles which intersect the region R. Thus we will have a = x_{0} < x_{1} <

where m
Definition 24.1.2 Let R be a bounded region in the xy plane and let f be a bounded function defined on R. We say f is Riemannn integrable if there exists a number, denoted by ∫ _{R}fdA and called the Riemannn integral such that if ε > 0 is given, then whenever one imposes a sufficiently fine mesh enclosing R and considers the finitely many rectangles which intersect R, numbered as

It is ∫ _{R}fdA which is of interest. The iterated integral should always be considered as a tool for computing this number. When this is kept in mind, things become less confusing. Also, it is helpful to consider ∫ _{R}fdA as a kind of a glorified sum. It means to take the value of f at a point and multiply by a little chunk of area dA and then add these together, hence the integral sign which is really just an elongated symbol for a sum.
The careful explanation of these ideas is contained later in a special chapter devoted to the theory of the Riemannn integral. It is not for the faint of heart. It is only there for those who have a compelling need to understand all the details. I have given proofs for everything substantive in this book, and this is no exception. However, it is much better and easier to do the integral as a Lebesgue integral.
Example 24.1.3 Let f
From the above discussion,

The reason for this is that x goes from 0 to 4 and for each fixed x between 0 and 4, y goes from 0 to the slanted line, y = x, the function being defined to be 0 for larger y. Thus y goes from 0 to x. This explains the inside integral. Now ∫ _{0}^{x}

What of integration in a different order? Lets put the integral with respect to y on the outside and the integral with respect to x on the inside. Then

For each y between 0 and 4, the variable x, goes from y to 4.

Now

Here is a similar example.
Example 24.1.4 Let f
Put the integral with respect to x on the outside first. Then

because for each x ∈

and so

Now do the integral in the other order. Here the integral with respect to y will be on the outside. What are the limits of this integral? Look at the triangle and note that x goes from 0 to 4 and so 2x = y goes from 0 to 8. Now for fixed y between 0 and 8, where does x go? It goes from the x coordinate on the line y = 2x which corresponds to this y to 4. What is the x coordinate on this line which goes with y? It is x = y∕2. Therefore, the iterated integral is

Now

and so

the same answer.
A few observations are in order here. In finding ∫ _{S}fdA there is no problem in setting things up if S is a rectangle. However, if S is not a rectangle, the procedure always is agonizing. A good rule of thumb is that if what you do is easy it will be wrong. There are no shortcuts! There are no quick fixes which require no thought! Pain and suffering is inevitable and you must not expect it to be otherwise. Always draw a picture and then begin agonizing over the correct limits. Even when you are careful you will make lots of mistakes until you get used to the process.
Sometimes an integral can be evaluated in one order but not in another.
Example 24.1.5 For R as shown below, find ∫ _{R} sin
Setting this up to have the integral with respect to y on the inside yields

Unfortunately, there is no antiderivative in terms of elementary functions for sin

and ∫ _{0}^{8}

This illustrates an important idea. The integral ∫ _{R} sin