Chapter 5: Double Integration
Section 5.1 - The Double Integral
Section 5.2 - Iterated Double Integrals
Section 5.3 - Regions with Curved Boundaries
Section 5.4 - Changing the Order of Iteration
Section 5.5 - Numeric Evaluation of Iterated Integrals
Section 5.6 - Changing Variables in a Double Integral
Section 5.7 - Double Integration in Polar Coordinates
In single-variable calculus, the definite integral is defined as the limit of a Riemann sum, that is, of a sum of the areas of increasingly more, and narrower, rectangles that fit under the graph of a function. Similarly, for functions of two variables, the definite integral, called the double integral, is defined as the limit of a Riemann sum, now the sum of volumes of parallelepipeds that fit under the surface that is the graph of the function. The limit must be the bivariate limit, not an iterated limit. However, it turns out that for "nice" integrands, the double integral can be evaluated by one of two different iterated single integrals.
Numeric techniques for evaluating double integrals are inspired by methods that work for single integrals, but are necessarily more intricate. Maple has built-in functionality for such evaluations; deeper study of these techniques is typically reserved for courses in numerical analysis.
Coordinate changes in a double integral require functional substitutions in the integrand, a redefinition of the limits of integration in the new coordinate system, and the inclusion of the absolute value of the Jacobian of the transformation to the new system. The example studied in this chapter is the change from Cartesian to polar coordinates.
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