13.1: Iterated Integrals and Area

In the previous chapter we found that we could differentiate functions of several variables with respect to one variable, while treating all the other variables as constants or coefficients. We can integrate functions of several variables in a similar way. For instance, if we are told that , we can treat as staying constant and integrate to obtain :

Make a careful note about the constant of integration, . This "constant'' is something with a derivative of with respect to , so it could be any expression that contains only constants and functions of . For instance, if , then . To signify that is actually a function of , we write:

Using this process we can even evaluate definite integrals.

Example : Integrating functions of more than one variable

Evaluate the integral

Solution

We find the indefinite integral as before, then apply the Fundamental Theorem of Calculus to evaluate the definite integral:

We can also integrate with respect to . In general,

and

Note that when integrating with respect to , the bounds are functions of (of the form and ) and the final result is also a function of . When integrating with respect to , the bounds are functions of (of the form and ) and the final result is a function of . Another example will help us understand this.

Example : Integrating functions of more than one variable

Evaluate .

Solution

We consider as staying constant and integrate with respect to :

Note how the bounds of the integral are from to and that the final answer is a function of .

In the previous example, we integrated a function with respect to and ended up with a function of . We can integrate this as well. This process is known as iterated integration, or multiple integration.

Example : Integrating an integral

Evaluate

Solution

We follow a standard "order of operations'' and perform the operations inside parentheses first (which is the integral evaluated in Example .)

Note how the bounds of were to and the final result was a number.

The previous example showed how we could perform something called an iterated integral; we do not yet know why we would be interested in doing so nor what the result, such as the number , means. Before we investigate these questions, we offer some definitions.

Definition: Iterated Integration

Iterated integration is the process of repeatedly integrating the results of previous integrations. Integrating one integral is denoted as follows.

Let , , and be numbers and let , , and be functions of and , respectively. Then:

Again make note of the bounds of these iterated integrals.

With , varies from to , whereas varies from to . That is, the bounds of are curves, the curves and , whereas the bounds of are constants, and . It is useful to remember that when setting up and evaluating such iterated integrals, we integrate "from curve to curve, then from point to point.''

We now begin to investigate why we are interested in iterated integrals and what they mean.

Area of a plane region

Consider the plane region bounded by and , shown in Figure . We learned in Section 7.1 (in Calculus I) that the area of is given by

Figure : Calculating the area of a plane region R with an iterated integral.

We can view the expression as

meaning we can express the area of as an iterated integral:

In short: a certain iterated integral can be viewed as giving the area of a plane region.

A region could also be defined by and , as shown in Figure . Using a process similar to that above, we have

Figure : Calculating the area of a plane region R with an iterated integral.

We state this formally in a theorem.

THEOREM : Area of a plane region

1. Let be a plane region bounded by and , where and are continuous functions on . The area of is
2. Let be a plane region bounded by and , where and are continuous functions on . The area of is

The following examples should help us understand this theorem.

Example : Area of a rectangle

Find the area of the rectangle with corners and , as shown in Figure .

Figure : Calculating the area of a rectangle with an iterated integral in Example .

Solution

Multiple integration is obviously overkill in this situation, but we proceed to establish its use.

The region is bounded by , , and . Choosing to integrate with respect to first, we have

We could also integrate with respect to first, giving:

Clearly there are simpler ways to find this area, but it is interesting to note that this method works.

Example : Area of a triangle

Find the area of the triangle with vertices at , and , as shown in Figure .

Figure : Calculating the area of a triangle with iterated integrals in Example .

Solution

The triangle is bounded by the lines as shown in the figure. Choosing to integrate with respect to first gives that is bounded by to , while is bounded by to . (Recall that since -values increase from left to right, the leftmost curve, , is the lower bound and the rightmost curve, , is the upper bound.) The area is

We can also find the area by integrating with respect to first. In this situation, though, we have two functions that act as the lower bound for the region , and . This requires us to use two iterated integrals. Note how the -bounds are different for each integral:

As expected, we get the same answer both ways.

Example : Area of a plane region

Find the area of the region enclosed by and , as shown in Figure .

Figure : Calculating the area of a plane region with iterated integrals in Example .

Solution

Once again we'll find the area of the region using both orders of integration.

Using :

Using :

Changing Order of Integration

In each of the previous examples, we have been given a region and found the bounds needed to find the area of using both orders of integration. We integrated using both orders of integration to demonstrate their equality.

We now approach the skill of describing a region using both orders of integration from a different perspective. Instead of starting with a region and creating iterated integrals, we will start with an iterated integral and rewrite it in the other integration order. To do so, we'll need to understand the region over which we are integrating.

The simplest of all cases is when both integrals are bound by constants. The region described by these bounds is a rectangle (see Example ), and so:

When the inner integral's bounds are not constants, it is generally very useful to sketch the bounds to determine what the region we are integrating over looks like. From the sketch we can then rewrite the integral with the other order of integration.

Examples will help us develop this skill.

Example : Changing the order of integration

Rewrite the iterated integral with the order of integration .

Solution

We need to use the bounds of integration to determine the region we are integrating over.

The bounds tell us that is bounded by and ; is bounded by 0 and 6. We plot these four curves: , , and to find the region described by the bounds. Figure shows these curves, indicating that is a triangle.

Figure : Sketching the region R described by the iterated integral in Example .

To change the order of integration, we need to consider the curves that bound the -values. We see that the lower bound is and the upper bound is . The bounds on are to . Thus we can rewrite the integral as

Example : Changing the order of integration

Change the order of integration of .

Solution

We sketch the region described by the bounds to help us change the integration order. is bounded below and above (i.e., to the left and right) by and respectively, and is bounded between 0 and 4. Graphing the previous curves, we find the region to be that shown in Figure .

Figure : Drawing the region determined by the bounds of integration in Example .

To change the order of integration, we need to establish curves that bound . The figure makes it clear that there are two lower bounds for : on , and on . Thus we need two double integrals. The upper bound for each is . Thus we have

This section has introduced a new concept, the iterated integral. We developed one application for iterated integration: area between curves. However, this is not new, for we already know how to find areas bounded by curves.

In the next section we apply iterated integration to solve problems we currently do not know how to handle. The "real" goal of this section was not to learn a new way of computing area. Rather, our goal was to learn how to define a region in the plane using the bounds of an iterated integral. That skill is very important in the following sections.