Week 9: Double integrals, Areas, Moments and centers of mass, Double integrals in polar form

Thomas Calculus, 11th edition, pages: 1067 - 1091

Chapter 15 Multiple Integrals

In this chapter we consider the integral of a function of two variables ƒ(x, y) over a region in the plane and the integral of a function of three variables ƒ(x, y, z) over a
region in space. These integrals are called multiple integrals and are defined as the limit of approximating Riemann sums, much like the single-variable integrals presented in Chapter 5. We can use multiple integrals to calculate quantities that vary over two or three dimensions, such as the total mass or the angular momentum of an object of varying density and the volumes of solids with general curved boundaries.

Sec 15.1 Double Integrals

In Chapter 5 we defined the definite integral of a continuous function ƒ(x) over an interval [a, b] as a limit of Riemann sums. In this section we extend this idea to define the integral of a continuous function of two variables ƒ(x, y) over a bounded region R in the plane. In both cases the integrals are limits of approximating Riemann sums. A similar method of partitioning, multiplying, and summing is used to construct double integrals, as was used to define the definite integral of a continuous function ƒ(x). However, this time we pack a planar region R with small rectangles, rather than small subintervals. We then take the product of each small rectangle’s area with the value of ƒ at a point inside that rectangle, and finally sum together all these products. When ƒ is continuous, these sums converge to a single number as each of the small rectangles shrinks in both width and height.
The limit is the double integral of ƒ over R.

Sec 15.2 Areras, Moments, and Centers of Mass

In this section, we show how to use double integrals to calculate the areas of bounded regions in the plane and to find the average value of a function of two variables. Then we study the physical problem of finding the center of mass of a thin plate covering a region in the plane.

Sec 15.3 Double Integrals in Polar Form

Integrals are sometimes easier to evaluate if we change to polar coordinates. This section shows how to accomplish the change and how to evaluate double integrals over regions whose boundaries are given by polar equations.