Iterated Integrals

This section introduces iterated integrals as an algebraic tool and motivates their use in evaluating double integrals by using a horizontal or vertical slicing of the domain. Because we are still using rectangular regions of integration, we do not yet introduce horizontally or vertically simple descriptions of regions in the plane. This section can likely be done in a single class but sets up many of the elements that will be defined and used in double integrals over non-rectangular regions. While this section can be given a quick description, the practice with the mechanics of slicing and setting up interated integrals to evaluate double integrals is an important step in student understanding.

In , we defined the double integral of a continuous function \(f = f(x,y)\) over a rectangle \(R = [a,b] \times [c,d]\) as \[\begin{aligned}\end{aligned}\] using the classic calculus approach. Thus \(\iint_R f(x,y) \, dA\) is a limit of double Riemann sums. However, while this definition tells us exactly what a double integral is, it is not very helpful for computing the value of a double integral. Fortunately, there is a way to view a double integral as an iterated integral, which will make computations feasible in many cases.

The viewpoint of an iterated integral is closely connected to an important idea from single-variable calculus. When we studied solids of revolution, such as the one shown in Figure, we saw that in some circumstances we could slice the solid perpendicular to an axis and approximate the volume using circular disks or washers. From there, we were able to find the volume of each disk, after which we used an integral to add the volumes of the slices and found the total volume of the rotated solid.

We will consider an example of this approach in the following Preview Activity. Specifically, you examine how to how use single-variable integrals along traces in a similar way to the rotation volume described above. We will generalize this approach in this section for double integrals over rectangular regions. In , we will look at using this technique on non-rectangular regions of integration.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

The ideas that we explored in Preview Activity work more generally and lead to the idea of an iterated integral. We will explore how these ideas generalize by considering a surface \(z=f(x,y)\) such as the one shown in over a rectangular region \(R\) satisfying \(a\leq x\leq b\) and \(c\leq x\leq d\). Our goal will be to use an interated integral to find the volume beneath this surface and above the \(xy\)-plane over the region \(R\).

To avoid technical issues, we assume that \(f\) is continuous. As in the Preview Activity, we define a function \[\begin{aligned}\end{aligned}\]. The function \(A(x)\) determines the net signed area of a slice of the solid. The slice has a fixed value of \(x\). By adjusting the slider in , you can see a tickened version of such a slice, which we will refer to as a slab. (Thus, a slice will is a two-dimensional object and a slab is a three-dimensional object.) You should also observe how the shape of the slice changes as the value of \(x\) changes. This illustrates how it is essential to use an integral to find the area of a cross section, as this area cannot be determined by using only the value of \(x\) and simple formulas from geometry.

If we add up the volumes of the slabs in , then we obtain an approximation for the volume of the solid between the surface and the \(xy\)-plane over the region \(R\). If we let \(\Delta x\) denote the thickness of each slab, the volume of the slab with constant \(x\)-value \(x_i^*\) is \(A(x_i^*)\Delta x\). Hence, the sum \[\begin{aligned}\end{aligned}\] is an approximation of this volume. This sum is a Riemann summ of the single-variable function \(A\). Therefore, as we let the number \(m\) of slabs go to infinity we obtain the integral \[\begin{aligned}\end{aligned}\] as the volume of the solid.

Now remember how we originally defined \(A(x)\) by using an integral. Thus, since the double integral represents the (net signed) volume of the solid enclosed by the \(xy\)-plane and the surface \(z=f(x,y)\) over the region \(R\), we have \[\begin{aligned}\end{aligned}\]. We call these nested integrals an iterated integral, which will prove to be a frequently-used tool for evaluating double integrals.

The fact that integrating in either order results in the same value is known as Fubini's Theorem.

If \(f = f(x,y)\) is a continuous function on a rectangle \(R = [a,b] \times [c,d]\), then \[\begin{aligned}\end{aligned}\]

This is another time in which we trade a multi-dimensional problem (double integral) for multiple one-dimensional problems that we have much more experience with.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.