M408M Learning Module Pages

Main page

Chapter 10: Parametric Equations and Polar Coordinates

Chapter 12: Vectors and the Geometry of Space


Chapter 13: Vector Functions

Chapter 14: Partial Derivatives

Chapter 15: Multiple Integrals

Learning module LM 15.1: Multiple integrals

       Background
      What is a double integral?
      Volumes are double integrals

Learning module LM 15.2: Multiple integrals over rectangles:

Learning module LM 15.3: Double integrals over general regions:

Learning module LM 15.4: Double integrals in polar coordinates:

Learning module LM 15.5a: Multiple integrals in physics:

Learning module LM 15.5b: Integrals in probability and statistics:

Learning module LM 15.10: Change of variables:

Background

Double Integrals I -- Background

When talking about any mathematical quantity, it's important to keep track of three different questions:

  • What is it?
  • How do you compute it?
  • What is it good for?

This is especially true for integration, where the three questions have very different answers:
  • The integral $\int_a^b f(x) dx$ is the limit of a sum: $\displaystyle{\lim_{n\to\infty} \sum_{i=1}^n f(x_i^*)\Delta x}$.
  • We compute integrals with the Fundamental Theorem of Calculus: $\int_a^b f(x) dx = F(b)-F(a)$, where $F'(x)=f(x)$.
  • We use integration to compute the total amount of stuff in the interval $[a,b]$, where stuff could be area under a curve, distance traveled, volume of a solid of revolution, arc length of a curve, or a host of other applications.


Of these applications, the easiest to understand is area under a curve, which is why we used area to introduce integrals. Likewise, we will use volume to introduce double integrals. But remember: ordinary integrals are about much more than area, and double integrals are about much more than volume.

In the following video, we review the ideas and definitions of one-dimensional integrals. Not how to compute an integral, but what it is and what it's good for. On the next page, we'll see how the exact same ideas work in two and three dimensions. We'll tackle 'how do you compute it?' in the next learning module.

Remember the main idea of integration: The whole is the sum of the parts. Integration is a procedure for computing bulk quantities like area, volume, mass, distance, moment of inertia, and wealth. If $f(x)$ is the density of a quantity on interest, then
  1. Break the interval $[a,b]$ into $N$ pieces, each of size $\Delta x = (b-a)/N$. Label the pieces by an index $i$ that goes from 1 to $N$.
  2. For each $i$, pick a sample point $x_i^*$ somewhere in the $i$-th interval.
  3. Approximate the amount of stuff in the $i$-th piece as $f(x_i^*) \Delta x$, where $x_i^*$ is an arbitrary point in the $i$-th interval.
  4. Approximate the total amount of stuff as $\displaystyle{\sum_{i=1}^N f(x_i^*) \Delta x}$.
  5. To get an exact answer, take a limit as $N \to \infty$:
$$\int_a^b f(x) dx = \lim_{N \to \infty} \sum_{i=1}^N f(x_i^*) \Delta x.$$


These ideas are illustated with the standard `area under a curve' example. The video on the right shows how this area can be approximated by adding up narrower and narrower rectangles, with $x_i^*$ being the left endpoint. (You may need to hit the 'replay' button a few times, since it's only 6 seconds long.)